People also used to think that establishing your dominance over a dog is the best way to get it to do your bidding, usually through physical abuse and aggression. People make new discoveries daily, and this is true of animal behaviour.
I don't want to villify your mother, or turn you against her, but it would be a shame to let an animal suffer because of beliefs that are no longer relevant. Just remember that you need to respect her as well, she isn't deliberately trying to upset you or hurt your budgie friend, she just has a different opinion based on a different experience to you. As
@macawpower58 said, she just needs time to catch up!