I think I'm in a unique position of having visited all four zoos in Group H within the last two years! I visited both Dallas and Chester both in 2018 actually. Due to this, I can supply species lists accurate to what I saw as of my last visit:
Dallas Zoo
Palm-Nut Vulture
African White-Backed Vulture
Northern Bald Eagle
African Fish-Eagle
American Harpy Eagle
Hooded Vulture
African Lappet-Faced Vulture
Canvasback
Blue-Winged Goose
West Indian Whistling-Duck
Fulvous Whistling-Duck
White-Faced Whistling-Duck
Marbled Teal
Orinoco Goose
Red-Crested Pochard
Northern Spur-Winged Goose
Southern Spur-Winged Goose (they keep a mixed pair)
New World Comb Duck
Northern Shoveler
Hottentot Teal
Red Shoveler
Common Shelduck
Southern Screamer
Trumpeter Hornbill
Wreathed Hornbill
Jackson's Hornbill
Green Wood-Hoopoe
Red-Legged Seriema
King Vulture
Andean Condor
Western White Stork
Maguari Stork
Saddle-Billed Stork
Marabou Stork
Wood Stork
Speckled Mousebird
Speckled Pigeon
Pied Imperial-Pigeon
Crested Pigeon
Blue-Bellied Roller
Vulturine Guineafowl
Helmeted Guineafowl
Blue Crane
Demoiselle Crane
White-Naped Crane
West African Black Crowned Crane
East African Grey Crowned Crane
Wattled Crane
Lady Ross's Turaco
Snowy-Crowned Robin-Chat
White-Headed Buffalo-Weaver
Taveta Golden Weaver
Amethyst Starling
Western Cattle Egret
Dalmatian Pelican
Eastern Brown Pelican
Great White Pelican
Lesser Flamingo
American Flamingo
Red-and-Yellow Barbet
Western Double-Eyed Fig-Parrot
African Penguin
Spectacled Owl
Total Taxa: 63
Chester Zoo
Cape Teal
White-Winged Wood Duck
Baer's Pochard
White-Faced Whistling-Duck
Old World Comb Duck
Hottentot Teal
Ruddy Shelduck
African White-Backed Duck
Great Indian Hornbill
Javan Rhinoceros Hornbill
Visayan Tarictic Hornbill
Sunda Wrinkled Hornbill
red-billed hornbill
Blacksmith Lapwing
Abdim's Stork
Black Stork
Blue-Naped Mousebird
Nicobar Pigeon
Grey-Capped Emerald Dove
Pied Imperial-Pigeon
Mindanao Bleeding-Heart Dove
Luzon Bleeding-Heart Dove
Cinnamon Ground-Dove
Victoria Crowned-Pigeon
White-Naped Pheasant-Pigeon
Green-Naped Pheasant-Pigeon
Black-Naped Fruit-Dove
Pink-Headed Fruit-Dove
Eastern Superb Fruit-Dove
Socorro Dove
Sunbittern
Red-Billed Curassow
Kenyan Crested Guineafowl
Helmeted Guineafowl
Congo Peafowl
Himalayan Monal
Salvadori's Pheasant
Mountain Peacock-Pheasant
Palawan Peacock-Pheasant
Cabot's Tragopan
West African Black Crowned Crane
East African Grey Crowned Crane
Wattled Crane
White-Crested Turaco
Javan Green-Magpie
Asian Azure-Winged Magpie
Indian Red-Billed Blue Magpie
Timor Sparrow
Java Sparrow
Timor Zebra Finch
Montserrat Oriole
Asian Fairy-Bluebird
Sumatran Laughingthrush
Blue-Crowned Laughingthrush
Spotted Laughingthrush
Silver-Eared Mesia
Omei Shan Liocichla
Red-Tailed Laughingthrush
Snowy-Crowned Robin-Chat
White-Rumped Shama
Red Fody
Western Village Weaver
Asian Glossy Starling
Purple Glossy Starling
Bali Mynah
Red-Winged Starling
Grosbeak Starling
Yellow Cardinal
Scarlet-Rumped Tanager
Chestnut-Backed Thrush
Grey-Winged Blackbird
Kilimanjaro White-Eye
Little Egret
Hamerkop
American Flamingo
Black-Browed Barbet
Fiery-Tufted Barbet
Green Aracari
Hyacinth Macaw
Yellow-Backed Chattering Lory
Brazilian Grey-Breasted Conure
Humboldt Penguin
Luzon Lowland Scops-Owl
Northern White-Faced Owl
Spectacled Owl
Brown Wood-Owl
Ural Owl
Collared Trogon
Total Taxa: 88
Dallas' collection is better than I remembered it being, especially when it comes to birds-of-prey, waterfowl, and cranes. Chester still has the superior collection, though. I'm not sure about Dallas, but Chester's number will actually be higher than what I've reported due to them having several parrot species behind the scenes as well as some species I've missed on my visits that I couldn't remember exactly. I also left out some species I'm unclear of due to changes to the zoo since my last visit. It's worth noting that a good few of their animals are singletons and not really present for breeding, but they still do have some very choice breeding programs in the green-magpie and other more endangered species. In general they seem to have a rarer collection as well, despite Dallas having a few heavy hitters.
It should also be noted that Dallas' exhibitry for their birds is rather bland for the most part. Rather simply-- though nice-- aviaries scattered throughout the zoo, a large waterfowl pond, and a couple of walk-through aviaries. Chester on the other hand, while they do still have a few rather simply aviaries, has some truly amazing walk-through aviaries in Monsoon Forest (still counts imo..) and Tropical Realm.
In this case I'm voting 2-1 Chester and I can't really see much of an argument to chance my mind.
~Thylo