And tell yourself there's no such thing as super plastics. Why don't you discuss that with my Org. Chem. prof. HMM Why aren't self-cleaning ovens and toasters odorless? HMM Explain why as a QAQC did I test for polytetrafluoroethylene by smell first from the peanut butter factory cooker. HMM I guess that thesis I wrote at TAMU was phony.
I feel like I missed something in this thread. Hopefully this comment is useful for answering the OP's concerns - but I couldn't tell you - as my higher education never included organic chemistry.
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To OP - Firstly, congrats on your new little buddy, and for taking the time to do research into the best care for him/her. Have you posted photos of your bird anywhere? Sounds like a real cutie. Enjoy watching your little friend grow up!
For now, while you try to talk to your parents about non-stick (& various other acronym) pans, try to figure out when cooking will be happening. As others have suggested, you can close the door to the room, open (screened) windows as wide as weather permits, and make sure you're ventilating as well as possible.
Good luck! Keep doing research and asking questions.