Every airline has a different policy regarding pets in the cabin AND in cargo. If you do plan on bringing your parrot with you on a plane, make sure to check with the airline well in advance of your anticipated flight.
Here is an article on airline travel that you might find helpful:
ParrotChronicles.com
Keep in mind that most airlines that do allow pets in the cabin restrict the number of pets. If a passenger requires an assistance animal (ex. guide dog), that could result in a companion pet being bumped to cargo.
Some airlines don't allow pets AT ALL. Example:
Southwest Airlines Travel Policies - Animals and Pets
I had Pickle shipped to me in NJ by his breeder, located in FL. He flew from FL to GA and then to PA. There was an issue with the connecting flight, though. Pickle was taken off of the plane from FL but missed not one but 3 flights to PA. He seemed no worse for the wear when he was delivered to my house at 2am (after I waited over 4 hours at the airport). He stepped up from the carrier onto my hand, readily went into his cage and was eating like a champ the next day. In hindsight, I would have opted not to ship out on Saturday through GA, which is a very busy hub, apparently.
HTH!
-e-