You may need to get into the cage in case of an emergency or delay. Uncontrollable screaming would not be acceptable on a plane. The bird must be able to adapt easily to a new and changing environment without a phobic episode. There will be a need to take a car, bus or train from the plane to your destination, so it is helpful if the bird rides well in a vehicle . There will be crowds at the airport and perhaps at your destination.
People are curious about pets in carriers.
The bird must be able to stay in it's carrier throughout the flight and on public and even private transportation. The bird must be patient while in the carrier or while waiting in case of undue delays. Looking over the chart, the cockatoo or amazon would be the most likely candidates for air travel. It isn't only the parrot that needs to be able to tolerate the flight, but you that must be patient with the bird and willing to trouble shoot and quickly solve problems that might arise. Your activities might be possibly limited during your trip with your bird along.