Does one have the right or just the privilege to extend the PhD study period due to parental leave? Assuming somebody is employed as a PhD student, I found several references to PhD study period and parental leave, but the wording is rather weak.


  The study period may only be extended if there are special reasons for it. Such grounds may be due to illness, leave for military service, work for trade unions and student organizations or parental leave. Ordinance (2010:1064). [automatically translated, highlights mine]


Notice may instead of must.

The Regulations for doctoral studies at Umeå University states:


  5.4.  Parental    leave
  
  ● Parental leave shall always be reported to the employer in accordance with applicable law and agreements.
  
  ● When
  parental leave is cited as a special reason for extension of a
  doctoral studentship or term of a doctoral grant, the period of
  parental leave is counted from the day it is validated.
  
  The Dean makes the decisions on these matters.
  [highlights mine]


Notice that the Dean has the final word.

The PhD Student FAQ from the Stockholm University Student Union uses stronger wording:


  This means that you have the right to get prolongation for the time that you are on parental leave so that you don’t lose out of any time for your research.


Can anybody point me to a regulation, a legal precedent or anything similar clarifying whether prolongation of PhD study period is a right or a privilege?
The word 'may' does not preclude refusal. It means, it is possible to extend. 

In academia, in general, if there is good reason for extension of PhD, it is always allowed. Pregnancy is a good ground for extension and it would be very rare for a student to be penalised for getting pregnant.