I think you may have the location of the aviary mixed up. This photo is of the vulture aviary, which holds Eurasian black vulture, Northern bald ibis and Black swan. It indeed used to be walk-through a long while ago.
It is located near the Wallabies and (then) Flamingos, but it should be visible. You can actually see people looking into the aviary if you zoom in far enough.
There is a behind-the-scenes aviary that could have held Burrowing parrots at the time, but that's located more behind the Mangrove
