@Tim May When was this building torn down? I remember seeing a row of bird cages surrounding a building in 1997. There was a cage full of red-tailed black cockatoos. Was that this building?
This building still stands and still houses birds; it has now been renamed the Blackburn Pavilion.
(Incidentally, this building was built in 1888 as the Reptile House; its construction was partially financed by the sale of the famous African elephant "Jumbo". And manatees were once housed here in the floor pool originally intended for crocodilians.)
I think that the cockatoos you mention were kept in the parrot aviaries near the Three Island Pond.