Whether they have any extant function or not, they have lost their former function and in that sense they do fit the definition of vestigiality. Vestigial structures are often called vestigial organs, although many of them are not actually organs.
There are quite a few birds famous for doing ritualized mating dances, so I would only list 5: The Red Capped Manakin, The Black-footed albatross, The Western and Clark's grebe, The Superb bird of Paradise, & The Victoria's riflebird.
Parrots, such as African Greys and Amazons, aren't the only birds that can learn to imitate sounds, including human voices. Some other birds known for their ability to talk include crows, ravens, Indian Ringneck Parakeets, Budgies and Cockatiels.