Hi and welcome to Pettalk.

I don't know much about birds, but my aunt had a tiny parrot that was allergic to anything people use as cosmetics. He was also nervous and stressed. All that made him go itchy and in the end most of his feathers were gone (hence his name: E.T.).

I personally find it strange that your vet sends you home when there's obviously something wrong with the bird. Maybe you can go and see another vet?