Do Australia, New Zealand, Germany facilitate permanent stay visas to immigrant parents of citizens? If I get citizenship in either of the countries, can I bring my parents permanently to that country?
Australia has different visa options for parents wishing to join their citizen children permanently, depending on whether the applicant is in Australia or not at the time of the application.
Outside Australia: there are three possible visas. Two have extremely long waiting times (subclass 103 up to 30 years and subclass 114 up to 50 years), the other a high fee (subclass 143 AUD47,755).
Within Australia: five visa types are possible, depending on the circumstances.
To answer for New Zealand, it’s necessary to know your parents’ citizenship https://www.immigration.govt.nz/new-zealand-visas/options/live-permanently/explore-visa-options-for-living-permanently
In Germany, parents of adult residents and/or (EU-)citizens can only be considered for family reunification (Familiennachzug) if you can show they are dependent and need you to take care of them. So I think the answer is: no, having a child who is a resident or citizen doesn't confer any advantage. That's quite typical for European countries.