Would it be a good thing to put a mirror in my parrot cage?
No, do not put a mirror in its cage. It will try to bond with the bird in the mirror and will NOT bond with you. Unless it bonds with you, it will not talk. Large parrots learn fast, but smaller ones such as parakeets and cockatiels take longer (I think)! I taught my birds (smaller ones) to talk by making a recording of the phrases I wanted it to learn, repeating them over and over, as much as possible with the same intonation I wanted the bird to learn. Every time I left home, I would start playing it. Eventually the battery would go dead or I’d come home and shut it off. Good luck!