The term you want to research online for is "TCN"/"third country nationals". The rights of family members of TCNs is generally not managed at an EU wide level, but is left to the discretion of the EU member-states.
Page 6 of
this document from the Catholic University of Leuven may be a starting point to do your research.
Note that the the right of your adlt dependent parents to join you in an EU member-state may be conditional on them having adequate accomodation, taking part in integration and local language courses, proving that they have sufficient funds, etc, and that those conditions may vary from member-state to member-state.
If you are coming from a warm climate, you may want to choose a more southern member-state, like Spain or Italy, to start your research with. Italy has the advantage that Sicily is wonderfully warm and picturesque, whereas Bolzano, near the border with Austria, is heavily Austrian in culture, with German an official language and a strong German presence (it used to be a part of Austria till about 100 years ago). So you could tick both boxes (warm climate and proximity to Austria).
You may also want to factor in how long it would take
you, the main applicant, to acquire the citizenship of that member-state. I think
Austrian citizenship by naturalisation requires 10 years of residence in Austria, for instance, whereas Italy allows Italian citizenship after only three years if married to an Italian citizen and only a year and a half if also parents to an Italian child.
Once you become an EU citizen, your rights re dependent parents change considerably.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.