That's up to each air carrier. You should contact their customer service line to confirm. Personally I'd avoid shipping a pet in cargo.
We used United Pet Safe when moving across country. They did fine. The cat went in-cabin with me since he was small enough. Each airline is different. Southwest allows in-cabin, but they do not have cargo for pets. Parrots are very sensitive to their surroundings and temps, so I would definitely find an airline that allowed it to be in-cabin with you.
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