Q:

Does parsley help in cleansing the kidneys?

A quick Google with "Parsley Kidney Cleanse", would return a search results of blogs and personal articles where it claims how regularly drinking refrigerated filtered water boiled with Parsley helps in cleaning Kidney from accumulated Salt.
Some of these claims like
How To Clean Your Kidneys | Kidney Cleanse

Kidney Cleanse: Second Preparation | Ingredients:

4 bunches of fresh parsley

Rinse with very hot water to cleanse it. Add the parsley to 1 quart of
  water and boil for 5 minutes. Drink 1/4 cup as soon as it is cool
  enough. Refrigerate a pint and freeze 1 pint. Drink daily for 3 weeks:
  3/4 cup of the root mixture and 1/2 cup of the parsley water

Simple Kidney Cleanse With Parsley

Easy Steps For Kidney Cleanse.
Follow these easy steps for kidney cleanse.

Cut a bunch of parsley into smaller pieces.
Boil the parsley in a pot of water for 10 minutes.
Let it cool down and then sieve and discard the parsley.
Pour the filtered water into bottles or any container and keep in fridge.
Drink one glass of the water daily.
You should notice the sediments in your urine.
You should feel a sense of well-being after that.

Does Parsley Clean the Kidneys?

What's In It?
Parsley owes its diuretic effect to the presence of two ingredients,
  apiol and myristicin, which are found in parsley oil. The amount of
  these substances varies across the more than 30 varieties of parsley,
  and the amount of oil varies in different parts of the plant, being
  more concentrated in the leaves than in the root. The German agency
  Commission E indicates that parsley may be used for ridding the body
  of kidney stones. There is currently no clinical evidence to support
  this, according to the Tufts Medical Center, but increased urine flow
  could potentially have this effect.

Unfortunately none of the links leads me to a research paper or the claim that was backed by a reputed medical practitioner.
So, is this a Myth or a Fact?

A:

The sources from the OPs question are ‘naturally’ (and) dubious. Both in their presentation and claims.
For ‘cleaning a kidney’ a human body needs adequate amounts of water. Calling a concept relating to internal medicine a “cleansing” seems like a surefire way to indicate quackery. If the goal is to keep the kidneys in a good, healthy, working condition then parsley does not feature on a reputable list. Quite the opposite:

What about herbal supplements that act like a "water pill"? –– Some
herbal supplements that act like a diuretic or "water pill" may cause
"kidney irritation" or damage. These include bucha leaves and juniper
berries. Uva Ursi and parsley capsules may also have bad side effects.

This part from the livestrong link is beyond the borderline of absurd:

"The German agency Commission E indicates that parsley may be used for
ridding the body of kidney stones. There is currently no clinical
evidence to support this, according to the Tufts Medical Center[…]"

That commission was set up to promote clinical evidence and I haven't found anything that brings parsley and kidney stones together there. Further: using a diuretic to ‘get rid of’ a kidney stone will in the vast majority of cases just not work.
The original Commission E papers on parsley (in German and as images!):
Petroselini fructus (Petersilienfrüchte). /
Petroselini herba/-radix (Petersilienkraut/-wurzel).
Although there is a kernel of truth at the root.
One notable journal has the following claims substantiated:
Potency | Latin name           | Part used | Family   | Miscellaneous     | Commission E 
        | (common name)        |           |          | notes             | approved for diuresis
Strong  | Petroselinum crispum | Radix,    | Apiaceae | Antispasmodic,    | Yes  (root only)
        | (parsley)            | fructus   |          | anti-inflammatory |

.

Petroselinum crispus (parsley)
The humble parsley has an ancient reputation as a diuretic. The parts
used are either the root or the fruit (often incorrectly termed the
seed). Detailed investigations of aqueous extracts of parsley fruit
extracts have been conducted in rats. Rats fed aqueous extracts had
significantly higher 24-h urine outputs compared to when they drank
regular water. Mechanistic investigations suggested parsley was
inhibiting Na+/K+-ATPase, primarily in the renal cortex, and thus
interfering primarily with potassium secretion. This appeared to cause
the increased urine output. Though these findings do not place parsley
in the same category as any existing class of synthetic diuretic
drugs, they do strongly suggest parsley has diuretic and not just
aquaretic activity.
Two groups of constituents are believed to be
responsible for many of parsley’s effects – terpenoids, particularly
apiol, and flavonoids, particularly apigenin. The volatile oil extract
of parsley has shown, like its close cousin Apium graveolens (celery)
and apiol in isolation, calcium channel blocking activity in vitro.
This helps explain the traditional use of parsley as a carminative, or
agent that relieves intestinal spasms. Apigenin from an
orally administered parsley extract has been associated with
antioxidant activity in humans. Apigenin also exerts
anti-inflammatories effects in vitro.
Clinical trials were not located
regarding the efficacy of parsley as a diuretic or any other
indication. However, it is approved by the German Commission E for use
as a diuretic as discussed above. Typically 2 g of root or fruit are
decocted in a covered vessel at low heat for 10–15 min in a cup of
water, and three such cups are drunk each day. A usual dose of
tincture is 2–4 ml three times daily. Parsley should be avoided in
large doses in pregnancy as apiol may stimulate uterine contractions
and in renal failure or nephritis. Parsley contains furanocoumarins
that may cause photosensitivity though the quantities are so low the
chance of an actual problem from internal use of medicinal doses is
slight.

quoted from:
Eric Yarnell: Botanical medicines for the urinary tract.
World Journal of Urology
November 2002, Volume 20, Issue 5, pp 285–293
https://doi.org/10.1007/s00345-002-0293-0
(amply sourced references)
But that's about it: it is a herbal remedy of which certain parts have been used in the studies that can promote diuresis.
“Cleaning kidneys” and “drinking that regularly” are certainly not advised and also not approved by Commission E!
If it is intended to be used as medicine it should be done so under some clinical supervision – especially for long term use. (Which probably runs diametrically against using this herb in the first place?)
This is especially urgent advice since one of the side effects of using parsley may lead to what?: Renal damage!
According to the sources cited in Robert Tisserand's Essential Oil Safety Book there are also several genuine side-effects and numerous interactions to consider. And among the quotes from that book there is an updated commission E statement:

Interactions:

with MAOIs
opioid analgesics
SSRIs
indirect sympathomimetic drugs including ephedrine and amphetamine

Side effects:

Apiole is demonstrated potential for nephrotoxicity at therapeutic doses
In cases of fatal toxicity from parsley apiole ingestion, considerable liver damage is generally found post mortem
parsley leaf and seed oils, is thought to be the active constituent, and is also used in its own right as an abortifacient. Data
is often difficult to obtain, partly because of the legal implications of patients admitting to illegal abortion, and in some cases
because death followed abortion and there is no record of how much
apiole was taken.

Our safety advice: We have proposed a daily oral maximum for parsley
apiole of 0.4 mg/kg.
Reproductive toxicity: Apiole and various
preparations of parsley have been used for many years to procure
illegal abortion in Italy. Post-abortive vaginal bleeding, sometimes
profuse, is a feature of these cases. A cumulative effect is apparent,
parsley apiole being taken daily for between two and eight days before
either death or abortion ensued. The lowest daily dose of apiole which
induced abortion was 0.9 g taken for eight consecutive days.
Parsley apiole is toxic in humans; the lowest total dose of apiole
causing death is 4.2 g (2.1 g/day for 2 days) the lowest fatal daily
dose is 770 mg, which was taken for 14 days; the lowest single fatal
dose is 8 g. At least 19 g has been survived. Parsley apiole is
hepatotoxic and nephrotoxic.
Summary: In fatal or almost-fatal doses, parsley apiole is
abortifacient, and toxic to the liver, kidneys, heart and digestive
system. Safety thresholds have not been established.
Regulatory guidelines
The Commission E Monograph for parsleyseed oil
concludes that ‘a therapeutic application cannot be justified because
of high risks.’ (Blumenthal, M., Busse, W.R., Goldberg, A., et al.,
1998. The complete German Commission E monographs: therapeutic guide to herbal medicines. American Botanical Council, Austin, Texas.)

Another report that references the older Tisserand compilation but is available on the net:
https://hfnet.nih.go.jp/usr/kiso/ninpu-herb/HerbalsSafetyReportJuly2002_Final.pdf

