This has been a bug ever since tamable pets were added. I can remember 3 times when I accidentally killed my parrots. Because of this, I decided to go in creative mode to replace them. If you can't fix this bug, then parrots and cats should gain 20 health points when you tame it just like a dog, and giving them food should heal them like dogs. before the second time I accidentally killed my parrot, a witch threw a poison potion at me and some of my pets, and I should have been able to heal them by giving them food just like a dog.