user-1: 
In 2012, Harry Falk and Seishi Karashima published a damaged and partial Kharoṣṭhī manuscript of the Mahāyāna Aṣṭasāhasrikā Prajñāpāramitā Sūtra. It is carbon dated to ca. 75 CE, making it one of the oldest Buddhist texts in existence.
Wikipedia: Gandhāran Buddhist texts - The "Split" Collection

Here's a copy of that paper by Harry Falk and Seishi Karashima: A first-century Prajñāpāramitā manuscript from Gandhāra
Question (1): What are the commonly acceptable hypotheses/theories among Buddhist-studies scholars in order to explain how come Prajñāpāramitā sutra (which belongs to the so-called 'Mahayana') have existed since so early (no later than 75 CE)?
Question (2): Is it sensible to hypothesize/theorize (or even accept) that, Prajñāpāramitā thoughts, in fact, have developed early parallel with thoughts in Āgama and Nikāya (which belongs to the so-called 'Hinayana')?

user254: I'm not sure I understand the question.
What are some theories?
There are some theories about the origins of Mahayana introduced here on Wikipedia: Mahayana -- History
so early (no later than 75 CE)?
As for dates, maybe the dating is inline with what scholars expect -- for example Stanford scholar discusses Buddhism and its origins says,

As far as we know, Mahayana Buddhism began to take shape in the first century BCE. This religious movement then rapidly developed in a number of different places in and around what is now India, the birthplace of Buddhism.
Buddhism itself started sometime in the fifth century BCE. We now think that the Buddha, who founded the religion, died sometime toward the year 400 BCE. As Buddhism developed, it spread beyond India. A number of different schools emerged. And out of that already complicated situation, we had the rise of a number of currents, or ways of thinking, which eventually started being labeled as Mahayana.


user471: I quote all of these from the book "A History of Indian Buddhism: From Sakyamuni to Early Mahayana" (1990) by Hirakawa Akira (professor and widely respected academic authority in Buddhist Studies), and translated by Paul Groner. The whole book in PDF format can be downloaded here and here.
Firstly, the lifetime of the Buddha:

One of the most widely accepted theories is based on the Sri Lankan
historical chronicles, the Dipavamsa and the Mahavamsa. On the basis
of these sources, Wilhelm Geiger calculated that the Buddha died in
483 B.C.E. and consequently had been born in 563 B.C.E. Hermann
Jacobi, using the same method and sources, maintained that the Buddha
died in 484 B.C.E. The Japanese scholar Kanakura Ensho has arrived
at the same date. The "dotted record" transmitted along with the
Chinese translation of the Theravada commentary on the Vinaya, the
Samantapasadika (T 1462), also indicates a similar date.

So, the period called Early Buddhism started on 484 B.C.E. after the death of the historical Buddha.
What are the other periods?

Indian Buddhism may be divided into the following five periods: (1)
Early Buddhism, (2) Nikaya or Sectarian (often called Hinayana)
Buddhism, (3) early Mahayana Buddhism, (4) later Mahayana Buddhism, and
(5) Esoteric Buddhism. Although the five periods are arranged in the
chronological order in which the traditions arose, they are also based
on a categorization of types of Buddhism as much as historical
criteria.

When did Nikaya or Sectarian Buddhism start? According to the statement below, it started around 384 B.C.E. i.e. one century after the Buddha's death.

Approximately one century after the Buddha's death, the early order
split into the Mahasanghika and Sthaviravada schools. Later, further
schisms occurred, resulting in a number of additional schools. The
second period of Buddhist history is concerned with the development
of Sectarian (Nikaya) Buddhism. ....
The term "Nikaya Buddhism" refers to monastic Buddhism after the
initial schism into the Mahasanghika and Sthavira schools had occurred.

Mahasanghika was basically a liberal school and Sthaviravada was a conservative school. And this first schism was more related to the monastic rules (vinaya) rather than doctrines. Doctrinal differences developed later. This split begins the period of Nikaya or Sectarian Buddhism.

The schism, often called the basic schism ...,
resulted in the formation of two schools: the Mahasanghika, whose
monks refused to accept the conservative ruling of the committee of
eight monks, and the Sthaviravada (P. Theravada), whose monks agreed
with the conservative ruling. ....
At this time, the controversy over the ten points of vinaya arose, and
the elders met in Vaisali to deliberate over the disputes and resolve
them. Many monks did not submit to the council's decision, however,
and the dispute later became a cause for the schism that resulted in
the Sthavira and Mahasanghika schools.

It must be made clear that Mahayana is not Mahasanghika.

On the basis of such evidence, some scholars have argued that Mahayana
Buddhism might have developed out of the Mahasanghika School. In
fact, some connection seems to have existed between the two forms of
Buddhism. However, since the doctrines of the Mahasanghika School
and the schools that split off from it (such as the Purvasaila,
Uttarasaila, and Caitika) are not clearly known, the similarities
between Mahayana Buddhism and the schools in the Mahasanghika lineage
cannot be determined with precision.

When did early Mahayana Buddhism begin? It began when the first Mahayana texts started appearing.
Mahayana texts originated 500 years after the Buddha's death, and therefore, 400 years after the beginning of Nikaya or Sectarian Buddhism. This was around first century B.C.E.

The thousand years following the Buddha's death are often divided into
two five-hundred-year periods in Buddhist texts. Statements about the
decline of the true teaching during the latter five hundred years
occur frequently in Mahayana texts. The phrase "latter five hundred
years" is contrasted with the "former five hundred years," the first
five hundred years after the Buddha's death. ....
Mahayana texts stress that the true teaching had to be carefully
guarded and maintained during the latter five hundred years .. The
presence of such words in Mahayana texts suggests that these texts
were composed sometime later than five hundred years after the
Buddha's death.
Early Mahayana texts date from the first century B.C.E. If the Buddha
died in 484 B.C.E., then "the former five hundred years" would have
elapsed in the first century C.E. ... These dates must be reconciled
with the evidence suggesting that Mahayana texts began appearing in
the first century B.C.E.

What is the source of the doctrines of Mahayana Buddhism? It basically derived from multiple schools of Nikaya or Sectarian Buddhism. And again, Mahayana is not Mahasanghika.

Many
modern scholars have maintained the view that Mahayana Buddhism
developed out of the Mahasanghika School. But since the Mahasanghika
School continued to exist long after Mahayana Buddhism arose, the rise
of Mahayana cannot be explained simply as the transformation of the
Mahasanghika into Mahayanists. While it is true that the many
similarities between Mahasanghika and Mahayana doctrines prove that the
Mahasanghika School did influence Mahayana Buddhism, teachings from
the Sarvastivadin, Mahisasaka, Dharmaguptaka, and Theravada schools
were also incorporated into Mahayana Buddhism. The doctrines of the
Sarvastivada School in particular were often mentioned in Mahayana
texts, and Sammatiya teachings also were influential. The relation
between Nikaya Buddhism and Mahayana Buddhism clearly is not a simple
one.

However, Mahayana concepts have a strong connection to Mahasanghika.

According to the Samaya, Mahasanghika doctrine included certain views
on the bodies of the Buddha and the concept of the bodhsattva that
might have drawn opposition from more conservative monks. However,
these doctrines were probably developed by later Mahasanghika monks
and do not represent Mahasanghika doctrine at the time of the basic
schism. ....
In one of the sections of his work, Vasumitra grouped together the
doctrines of four schools (the Mahasanghika, Lokottaravadin,
Ekavyavaharika, and Kaukutika) of Mahasanghika lineage and noted
that the four taught that "the Buddhas; the World-honored Ones, are
all supermundane. All the Tathagatas are without impure (sasrava)
dharmas" (T49:15b). This position differs from that of the
Sarvastivadin School, but is close to Mahayana teachings.
These teachings are close to Mahayana ideas about the sambhogakaya
(body of bliss) of the Buddha and are evidence of the close
relationship of these schools to Mahayana Buddhism.

When did the first Mahayana text appear?
Apparently it was called Satparamita (six perfections), but it is no longer extant. And it appeared before the earliest extant Mahayana texts, which is the perfection of wisdom sutras, the Mahaprajnaparamitasutra. Satparamita was most probably compiled in the first century B.C.E.

The very earliest Mahayana scriptures such as the Satparamita are no
longer extant. Consequently, the date of their composition cannot be
determined from the texts themselves. However, approximate dates can
be determined indirectly. Early versions of texts such as the Kasyapa-
parivarta ... were probably compiled in the first century of the
common era. Since the Satparamita was quoted in these texts, the
Satparamita and the other earliest Mahayana texts were probably
compiled in the first century B.C.E. .... Perfection of wisdom sutras
probably first appeared after the Satparamita was compiled.

Now let's look at the perfection of wisdom sutras, which is dated to earliest first century C.E.

The largest perfection of wisdom text is the Tcr. pan-jo po-lo-mi-to
ching (T 220, Mahaprajnaparamitasutra) translated into Chinese by
Hsuan-tsang. It is six hundred fascicles long and divided into sixteen
assemblies (or parts). Perfection of wisdom sutras were not always
such large works. At first a number of separate texts circulated
independently. Later they were collected together to make larger works
such as the one mentioned above.
The oldest sutra in this group is the Tao-hsing pan-jo ching (T 224)
translated by Lokaksema. Since the translation was completed around
179, the original text probably dates back to the first century
C.E.

Now for the OP's questions:

OP: Question (1): What are the commonly acceptable hypotheses/theories
among Buddhist-studies scholars in order to explain how come
Prajñāpāramitā sutra (which belongs to the so-called 'Mahayana') have
existed since so early (no later than 75 CE)?

I think it's well explained above and in the book.

OP: Question (2): Is it sensible to hypothesize/theorize
(or even accept)
that, Prajñāpāramitā thoughts, in fact, have developed early parallel
with thoughts in Āgama and Nikāya (which belongs to the so-called
'Hinayana')?

Based on the quoted texts above and further details in the book, Mahayana doctrines definitely derived from multiple schools of Nikaya Buddhism, including Mahasanghika, Sarvastivadin, Mahisasaka, Dharmaguptaka, Theravada and Sammatiya (Pudgalavada). However, there are definitely extra strong links to schools which came out of Mahasanghika.
In conclusion, Mahayana and Prajnaparamita thoughts did not develop in parallel with Nikaya Buddhism, but was rather derived from it, and then developed further.
It's also interesting to note that the OP's dating of the perfection of wisdom sutras manuscript by Harry Falk and Seishi Karashima from 2012, and ChrisW's Stanford Buddhist scholar Paul Harrison's chronology comments from 2018, all match up nicely to Prof. Akira Hirakawa's chronology comments from 1990.

-1: <blockquote>
<p>In 2012, Harry Falk and Seishi Karashima published a damaged and partial Kharoṣṭhī manuscript of the Mahāyāna Aṣṭasāhasrikā Prajñāpāramitā Sūtra. It is carbon dated to ca. 75 CE, making it one of the oldest Buddhist texts in existence.</p>
<p><a href="https://en.wikipedia.org/wiki/Gandh%C4%81ran_Buddhist_texts#The_%22Split%22_Collection" rel="nofollow noreferrer">Wikipedia: Gandhāran Buddhist texts - The &quot;Split&quot; Collection</a></p>
</blockquote>
<p>Here's a copy of that paper by Harry Falk and Seishi Karashima: <a href="https://docdro.id/PyJF9z5" rel="nofollow noreferrer">A first-century Prajñāpāramitā manuscript from Gandhāra</a></p>
<p>Question (1): What are the commonly acceptable hypotheses/theories among Buddhist-studies scholars in order to explain how come Prajñāpāramitā sutra (which belongs to the so-called 'Mahayana') have existed since so early (no later than 75 CE)?</p>
<p>Question (2): Is it sensible to hypothesize/theorize (or even accept) that, Prajñāpāramitā thoughts, in fact, have developed early parallel with thoughts in Āgama and Nikāya (which belongs to the so-called 'Hinayana')?</p>
 254: <p>I'm not sure I understand the question.</p>
<p><strong>What are some theories?</strong></p>
<p>There are some theories about the origins of Mahayana introduced here on Wikipedia: <a href="https://en.wikipedia.org/wiki/Mahayana#History" rel="nofollow noreferrer">Mahayana -- History</a></p>
<p><strong>so early (no later than 75 CE)?</strong></p>
<p>As for dates, maybe the dating is inline with what scholars expect -- for example <a href="https://news.stanford.edu/2018/08/20/stanford-scholar-discusses-buddhism-origins/" rel="nofollow noreferrer">Stanford scholar discusses Buddhism and its origins</a> says,</p>
<blockquote>
<p>As far as we know, Mahayana Buddhism began to take shape <strong>in the first century BCE</strong>. This religious movement then rapidly developed in a number of different places in and around what is now India, the birthplace of Buddhism.</p>
<p>Buddhism itself started sometime <strong>in the fifth century BCE</strong>. We now think that the Buddha, who founded the religion, died sometime toward the year 400 BCE. As Buddhism developed, it spread beyond India. A number of different schools emerged. And out of that already complicated situation, we had the rise of a number of currents, or ways of thinking, which eventually started being labeled as Mahayana.</p>
</blockquote>
 471: <p>I quote all of these from the book &quot;<a href="http://hdl.handle.net/10125/23030" rel="nofollow noreferrer">A History of Indian Buddhism: From Sakyamuni to Early Mahayana</a>&quot; (1990) by <a href="https://buddhanature.tsadra.org/index.php/People/Hirakawa,_A." rel="nofollow noreferrer">Hirakawa Akira</a> (professor and widely respected academic authority in Buddhist Studies), and translated by Paul Groner. The whole book in PDF format can be downloaded <a href="https://scholarspace.manoa.hawaii.edu/bitstream/10125/23030/1/%2314_Hirakawa.pdf" rel="nofollow noreferrer">here</a> and <a href="http://www.ahandfulofleaves.org/documents/A%20History%20of%20Indian%20Buddhism_From%20Sakyamuni%20to%20Early%20Mahayana_Akira.pdf" rel="nofollow noreferrer">here</a>.</p>
<p>Firstly, the lifetime of the Buddha:</p>
<blockquote>
<p>One of the most widely accepted theories is based on the Sri Lankan
historical chronicles, the Dipavamsa and the Mahavamsa. On the basis
of these sources, Wilhelm Geiger calculated that the <strong>Buddha died in
483 B.C.E. and consequently had been born in 563 B.C.E.</strong> Hermann
Jacobi, using the same method and sources, maintained that the Buddha
died in 484 B.C.E. The Japanese scholar Kanakura Ensho has arrived
at the same date. The &quot;dotted record&quot; transmitted along with the
Chinese translation of the Theravada commentary on the Vinaya, the
Samantapasadika (T 1462), also indicates a similar date.</p>
</blockquote>
<p>So, the period called Early Buddhism started on 484 B.C.E. after the death of the historical Buddha.</p>
<p>What are the other periods?</p>
<blockquote>
<p>Indian Buddhism may be divided into the following five periods: (1)
Early Buddhism, (2) Nikaya or Sectarian (often called Hinayana)
Buddhism, (3) early Mahayana Buddhism, (4) later Mahayana Buddhism, and
(5) Esoteric Buddhism. Although the five periods are arranged in the
<strong>chronological order</strong> in which the traditions arose, they are also based
on a categorization of types of Buddhism as much as historical
criteria.</p>
</blockquote>
<p>When did Nikaya or Sectarian Buddhism start? According to the statement below, it started around 384 B.C.E. i.e. one century after the Buddha's death.</p>
<blockquote>
<p>Approximately <strong>one century after the Buddha's death, the early order
split</strong> into the Mahasanghika and Sthaviravada schools. Later, further
schisms occurred, resulting in a number of additional schools. The
second period of Buddhist history is concerned with the development
of Sectarian (Nikaya) Buddhism. ....</p>
<p>The term &quot;Nikaya Buddhism&quot; refers to monastic Buddhism after the
initial schism into the Mahasanghika and Sthavira schools had occurred.</p>
</blockquote>
<p>Mahasanghika was basically a liberal school and Sthaviravada was a conservative school. And this first schism was more related to the monastic rules (vinaya) rather than doctrines. Doctrinal differences developed later. This split begins the period of Nikaya or Sectarian Buddhism.</p>
<blockquote>
<p>The schism, often called the basic schism ...,
resulted in the formation of two schools: the Mahasanghika, whose
monks refused to accept the conservative ruling of the committee of
eight monks, and the Sthaviravada (P. Theravada), whose monks agreed
with the conservative ruling. ....</p>
<p>At this time, the controversy over the ten points of <strong>vinaya</strong> arose, and
the elders met in Vaisali to deliberate over the disputes and resolve
them. Many monks did not submit to the council's decision, however,
and the dispute later became a cause for the schism that resulted in
the Sthavira and Mahasanghika schools.</p>
</blockquote>
<p>It must be made clear that Mahayana is not Mahasanghika.</p>
<blockquote>
<p>On the basis of such evidence, some scholars have argued that Mahayana
Buddhism might have developed out of the Mahasanghika School. In
fact, some connection seems to have existed between the two forms of
Buddhism. However, since the doctrines of the Mahasanghika School
and the schools that split off from it (such as the Purvasaila,
Uttarasaila, and Caitika) are not clearly known, <strong>the similarities
between Mahayana Buddhism and the schools in the Mahasanghika lineage
cannot be determined with precision.</strong></p>
</blockquote>
<p>When did early Mahayana Buddhism begin? It began when the first Mahayana texts started appearing.</p>
<p>Mahayana texts originated 500 years after the Buddha's death, and therefore, 400 years after the beginning of Nikaya or Sectarian Buddhism. This was around first century B.C.E.</p>
<blockquote>
<p>The thousand years following the Buddha's death are often divided into
two five-hundred-year periods in Buddhist texts. Statements about the
decline of the true teaching during the latter five hundred years
occur frequently in Mahayana texts. The phrase &quot;latter five hundred
years&quot; is contrasted with the &quot;former five hundred years,&quot; the first
five hundred years after the Buddha's death. ....</p>
<p>Mahayana texts stress that the true teaching had to be carefully
guarded and maintained during the latter five hundred years .. The
presence of such words in Mahayana texts suggests that these texts
were composed sometime later than five hundred years after the
Buddha's death.</p>
<p>Early Mahayana texts date from the first century B.C.E. If the Buddha
died in 484 B.C.E., then &quot;the former five hundred years&quot; would have
elapsed in the first century C.E. ... These dates must be reconciled
with the evidence suggesting that <strong>Mahayana texts began appearing in
the first century B.C.E.</strong></p>
</blockquote>
<p>What is the source of the doctrines of Mahayana Buddhism? It basically derived from multiple schools of Nikaya or Sectarian Buddhism. And again, Mahayana is not Mahasanghika.</p>
<blockquote>
<p>Many
modern scholars have maintained the view that Mahayana Buddhism
developed out of the Mahasanghika School. But since the <strong>Mahasanghika
School continued to exist long after Mahayana Buddhism arose</strong>, the rise
of Mahayana cannot be explained simply as the transformation of the
Mahasanghika into Mahayanists. <strong>While it is true that the many
similarities between Mahasanghika and Mahayana doctrines prove that the
Mahasanghika School did influence Mahayana Buddhism, teachings from
the Sarvastivadin, Mahisasaka, Dharmaguptaka, and Theravada schools
were also incorporated into Mahayana Buddhism.</strong> The doctrines of the
Sarvastivada School in particular were often mentioned in Mahayana
texts, and Sammatiya teachings also were influential. The relation
between Nikaya Buddhism and Mahayana Buddhism clearly is not a simple
one.</p>
</blockquote>
<p>However, Mahayana concepts have a strong connection to Mahasanghika.</p>
<blockquote>
<p>According to the Samaya, Mahasanghika doctrine included certain views
on the bodies of the Buddha and the concept of the bodhsattva that
might have drawn opposition from more conservative monks. However,
these doctrines were probably developed by later Mahasanghika monks
and do not represent Mahasanghika doctrine at the time of the basic
schism. ....</p>
<p>In one of the sections of his work, Vasumitra grouped together the
doctrines of <strong>four schools (the Mahasanghika, Lokottaravadin,
Ekavyavaharika, and Kaukutika) of Mahasanghika lineage</strong> and noted
that the four taught that &quot;the Buddhas; the World-honored Ones, are
all supermundane. All the Tathagatas are without impure (sasrava)
dharmas&quot; (T49:15b). This position differs from that of the
Sarvastivadin School, but is close to Mahayana teachings.</p>
<p>These teachings are close to Mahayana ideas about the sambhogakaya
(body of bliss) of the Buddha and are evidence of the <strong>close
relationship of these schools to Mahayana Buddhism.</strong></p>
</blockquote>
<p>When did the first Mahayana text appear?</p>
<p>Apparently it was called Satparamita (six perfections), but it is no longer extant. And it appeared before the earliest extant Mahayana texts, which is the perfection of wisdom sutras, the Mahaprajnaparamitasutra. Satparamita was most probably compiled in the first century B.C.E.</p>
<blockquote>
<p>The very earliest Mahayana scriptures such as the Satparamita are no
longer extant. Consequently, the date of their composition cannot be
determined from the texts themselves. However, approximate dates can
be determined indirectly. Early versions of texts such as the Kasyapa-
parivarta ... were probably compiled in the first century of the
common era. <strong>Since the Satparamita was quoted in these texts, the
Satparamita and the other earliest Mahayana texts were probably
compiled in the first century B.C.E.</strong> .... Perfection of wisdom sutras
probably first appeared after the Satparamita was compiled.</p>
</blockquote>
<p>Now let's look at the perfection of wisdom sutras, which is dated to earliest first century C.E.</p>
<blockquote>
<p>The largest perfection of wisdom text is the Tcr. pan-jo po-lo-mi-to
ching (T 220, Mahaprajnaparamitasutra) translated into Chinese by
Hsuan-tsang. It is six hundred fascicles long and divided into sixteen
assemblies (or parts). Perfection of wisdom sutras were not always
such large works. At first a number of separate texts circulated
independently. Later they were collected together to make larger works
such as the one mentioned above.</p>
<p>The oldest sutra in this group is the Tao-hsing pan-jo ching (T 224)
translated by Lokaksema. Since the translation was completed around
179, <strong>the original text probably dates back to the first century
C.E.</strong></p>
</blockquote>
<p>Now for the OP's questions:</p>
<blockquote>
<p><strong>OP:</strong> Question (1): What are the commonly acceptable hypotheses/theories
among Buddhist-studies scholars in order to explain how come
Prajñāpāramitā sutra (which belongs to the so-called 'Mahayana') have
existed since so early (no later than 75 CE)?</p>
</blockquote>
<p>I think it's well explained above and in the book.</p>
<blockquote>
<p><strong>OP:</strong> Question (2): Is it sensible to hypothesize/theorize
(or even accept)
that, Prajñāpāramitā thoughts, in fact, have developed early parallel
with thoughts in Āgama and Nikāya (which belongs to the so-called
'Hinayana')?</p>
</blockquote>
<p>Based on the quoted texts above and further details in the book, Mahayana doctrines definitely derived from multiple schools of Nikaya Buddhism, including Mahasanghika, Sarvastivadin, Mahisasaka, Dharmaguptaka, Theravada and Sammatiya (Pudgalavada). However, there are definitely extra strong links to schools which came out of Mahasanghika.</p>
<p>In conclusion, Mahayana and Prajnaparamita thoughts did not develop in parallel with Nikaya Buddhism, but was rather derived from it, and then developed further.</p>
<p>It's also interesting to note that the OP's dating of the perfection of wisdom sutras manuscript by Harry Falk and Seishi Karashima from 2012, and ChrisW's Stanford Buddhist scholar Paul Harrison's chronology comments from 2018, all match up nicely to Prof. Akira Hirakawa's chronology comments from 1990.</p>
