I didn't know if the title of the thread was referring to the presence of Australian birds solely in zoos or just in general, so I just included any birds that are commonly held privately, as well as those that are in public collections. Here are just a few that I came up with at the top of my head that weren't mentioned previously:
Pink-Eared Duck: Rare, only three holders; Central Park, Sedgwick County, and Sylvan Heights
Australasian Shoveler: Rare, only three holders; Minnesota, Sedgwick County, Sylvan Heights
Cape Barren Goose: Common; 11 holders
Chestnut Teal: Somewhat common, nine holders
Pacific Black Duck: Rare, only two holders; Sedgwick County, & Sylvan Heights
Plumed Whistling Duck; Somewhat uncommon; only 5 holders
Wandering Whistling Duck: Rare, only three holders; Dallas, Fort Worth, & Sylvan Heights
Green Pygmy Goose: Extremely rare, only one holder: Sylvan Heights
Brolga: Extremely rare, only 1 holder; International Crane Foundation
Straw-Necked Ibis: Somewhat uncommon, but a fair amount of holders
Silver Gull: Rare; Kansas City Zoo and a couple others are the only holders
Eastern Rosella: Common in captivity, as well as in the pet trade
Princess Parrot: Somewhat uncommon in zoological collections, but common in the pet trade
Diamond Dove: Extremely common in the pet trade
Wonga Pigeon: Uncommon; 4-5 holders
White-Breasted Woodswallow: Don't know the exact number of holders, but they seem to be around in prevalent numbers in quite a few collections from what I've seen
Australian Magpie: Bronx is the only holder I know of, are there more?
Double-Barred Finch: Common in the pet trade
Gouldian Finch: Extremely common in the pet trade
Zebra Finch: Extremely common in the pet trade