Its completely possible.
You'll need to create an animation of a parrot on a shoulder properly spaced for use on a UO avatar, create a gump of a parrot on the characters shoulder, create an itemID image of the parrot, then add all of those to your client files and set the proper tiledata. Then just write some quick code for a parrot item that is equippable. That last thing would only take like 3 minutes.