I'll give an overview of what I've found so far:
Fresno Chaffee Zoo has a South American aviary and an Australian aviary; I'm not sure exactly what birds are in their collection, but I know that the Australian aviary had multiple species of parrots and Victorian crowned pigeons recently. Other birds include flamingos, pelicans in the award-winning Sea Lion Cove exhibit, rhinoceros hornbill, Guam rail, and at least one not-so-common species, the bateleur eagle. There is also an unknown number of birds in the African complex. For conservation efforts, FCZ contributes to the Hornbill Nest Adoption Program.
Disney's Animal Kingdom has a sizable collection of African birds, and an aviary with over 50 species of Asian birds. Notable species include African jacana, northern carmine bee-eater (I remember reading that DAK was having good breeding success with them), go-away-bird, African hoopoe (really??), great blue turaco, pitta, crowned pigeon, Madagascar crested ibis, painted stork, blue-crowned laughingthrush, Guam kingfisher, hornbills, and crested coua. For conservation, the Disney Conservation Fund has contributed to programs for Siberian cranes, anti-poisoning efforts for vultures in Asia and Kenya, and (apparently) other programs for African birds. Additionally, one-third of Disney's land is set aside for wildlife, and is a major destination for migratory birds.
I'm putting my vote with Disney for now.