Why a pet bird instead of a dog or cat?

One quick answer would be probably you would not be allergic to birds, but to dogs or cats.
For hundreds of centuries people had birds for pets.

The Persian and Indian writers wrote about parrots and other birds around 3000 years ago. Ancient Chinese kept pheasants. The ancient Egyptians had zoos to put their pigeons in.

Mockingbirds were used by the Romans and parrots used by Ancient Greeks to tell them if people entered their homes. Ravens because of their ability to talk were a prize possession of the Romans. The Romans created beautiful aviaries for their birds.
They are given credit for introducing different varieties of birds to Great Britain and Europe.