Why Is There an Australian Cockatoo in This Italian Renaissance Painting?

If you look closely at the Madonna della Vittoria, a Renaissance painting created in 1496 and hanging in the Louvre, you might notice something peculiar. To the left, just above the Virgin Mary's head, sits a sulfur-crested cockatoo. These large white birds are common enough today in petshops and zoos, but in the 15th century, they were an exotic specimen of wildlife found primarily in New Guinea and Australia, the Guardian points out.
 How, exactly, did that cockatoo finagle its way into an Italian painting?