For the Carolina parakeet: perhaps have the pet trade go for captive breeding early on and have the general mood on them be either neutral or mixed. Also, moves made to protect the Ivory-billed Woodpecker could potentially help protect the Carolina parakeet too.
Either that, or have John Muir warn Theodore Roosevelt that the Carolina Parakeet, Passenger Pigeon and Ivory-billed Woodpecker (although I believe the Ivory-billed Woodpecker is actually still alive) are in danger of extinction early on. You could have the pair set up a program offering live specimens to be kept and bred in zoos.
You could also have at least the Carolina Parakeet or Ivory-billed Woodpecker be introduced (accidentally or intentionally) elsewhere. Maybe someone releases some Ivory-bills in places like California, New Zealand and Southern Australia (preferably Tasmania), where they ironically get themselves established and become an invasive species.
Really, you could save a lot of animals by introducing them elsewhere, as long as they can establish themselves in the new environment.