Before I even start, I warn you it is very difficult at best, impossible at worst. Despite what I write, you will come across many different obstacles. It seems no two cases are the same, certainly not the two cases I know of very well and heresy I've heard from others. You had better REALLY want your bird to be with you. In my case, there was no way on Earth I was moving without him, so I battled through all the red tape, and trust me, there's enough red- tape to stretch from where your leaving to where you're going!

I'll give you an account of the two cases I know. Both apply to travel to Belize.

Case 1. Dino, my Green-winged Macaw. IUCN listed as least concern. CITES II.
Traveling from Madrid, Spain to Belize City, Belize.

The very first thing you need is a CITES export license. Without that, there's no point in proceeding any further. I got that from Spain in just two weeks and it was valid for 6 months.
I then applied for the CITES import license from Belize. The CITES import permit should be nothing more than filling in a form, acknowledging that this is a CITES listed animal and is able to move internationally, but no. I received all manner of emails asking me questions about where I'd keep him, what I'd do to prevent his escape, what I'd do if he did escape, then long periods where I waited for replies to my answers to that, then more or less the same questions all over again, then more waiting and emailing back and forth. *Eventually it was issued but I also had to apply for the Forestry Department import license and for that, I had to have tests done to prove he was negative to West Nile virus, Newcastle disease and a very specific form of Avian flue. But its not that simple either. Had there have been an outbreak of any of those diseases in Spain in the last 6 months, the application would have been denied. Thankfully, there had been one single bird that had contracted Avian flue in spain 8 months earlier, but which would have been sufficient to deny the application. That therefore is something you should find out even before you apply for the CITES export permit. No point in wasting time and money on that if one of those diseases has been present in the country/region of origin because you simply will not be able to bring your bird to Belize.
Ok, if all's well there and you get the permits, 10 days before flying, your bird must get an International Certificate of Health from your vet. Without that, the bird will not be allowed to board a plane.
Seven days before flying, you need to contact the cargo department of the airline you are transporting the bird with to arrange to have a vet waiting at the airport to inspect the bird before he boards the plane. Again, he will not be permitted on the plane without the go ahead from the airport vet. You also need to contact BAHA in Belize to arrange for a vet to be waiting at the airport here to inspect the bird. Good luck with that, neither in my case nor the next case I'll write about was there a vet at the airport, despite it being arranged in advance. In this case, the head vet of BAHA gave the go ahead for Dino to enter via phone, saying he would come to our house the following day to check on him, which he did.
Within 3 days of flying, yet another vet visit to ensure the bird has no visible injuries of parasites, with an official letter of confirmation. If the bird has either, again, its a no go!

Well, I made it almost all the way there. I got as far as booking my flight with Continental Airlines who on numerous occasions had told me via email that birds could travel with them, only to find out from the booking clerk that since the plane had to touch down in the USA, they would not permit a bird on board, since none of the paperwork related to the USA but to Belize!!! This was just 10 days before I intended to fly. I was devastated. I'd got this far, now I'd come up against a hundred foot high brick wall at the last moment. By sheer luck, my friend in Belize had found a professional animal shipper in Madrid, Spain. I contacted the company, explained everything and they said they'd work on it. It involved them having to hire someone in the USA to take care of the plane swap, vet check there and paperwork. It was the only option left to me, so I took it, but it cost me $6,000 for their services. I just thank God I had the money. In all, it cost me $10,000 to get Dino here.

All done now? No, not likely! Fortunately Belize permits you to quarantine your bird in your own home, however a BAHA vet comes out roughly once a week, unannounced to check on the bird. Quarantine is one month and of course you foot the bill, as you do for every vet visit, license, form, travel etc etc.

*Having got the all clear from the vet regarding the diseases, I had to take the paperwork first to the College of  Veterinarians to have it certified, who then sent me to the Spanish health authority to certify them, who sent me to the Forestry Department to certify them!

My advise is, if you are traveling via another country, even if its only a stop over, seek the services of a professional. You simply won't be able to do it yourself.

Case 2. Rosie, my friend's Scarlet Macaw. IUCN listed as least concern. CITES II.
Traveling from Boston, USA to Belize City, Belize.

To cut another long story short, all of the above applies, except because the flight is direct, you will be able to take care of it yourself. However...

The fun begins! Jenni (my friend) who learned a lot from me shipping Dino attempted to do everything in the same order. But the problems she encountered were that after 5 months, the CITES export permit had still not been issued. At that point she started calling them daily, still to no avail, so in desperation resorted to calling every 10 minutes until they got so sick of her, they just wanted to get rid of her. But even at that point, they complicated the issue. They insisted the CITES import permit had to be issued before the export permit. (I was informed both by Spain and Belize that, that was the case up to a couple of years ago, but CITES changed it to the other way round - makes perfect sense, what's the point of applying for an import permit if you don't have the export permit?) But lo and behold, either the USA didn't know or just didn't want to. Fortunately, Jenni was able to convince both BAHA and CITES here to issue the permits, at which point the export permit was issued.

Jenni booked her flight, having taken care of everything else mentioned in Case 1 but on arrival at Belize City, the vet hadn't turned up and whereas in my case, he had already forwarded the landing permit to the airport so was able to ok Dino's passage via phone, had not done so for Rosie. Jenni was told by BAHA at the airport that she would have to come back at 10 am the next morning, leaving Rosie in her cage in the BAHA office overnight, but was also told that if the landing permit was not issued by 11 am Rosie would either be shipped back to Boston or put down. Can you imagine how Jenni felt at hearing that? (Had they have done that, we'd have caused an international outrage over it!). Anyway, the following morning the vet did turn up and Rosie was released.

I'm very happy to say, both Dino, who's trans-Atlantic trip took 4 days and Rosie's that was over a day, both weathered it far better than either of us had expected and that they are now living happily ever after with their very happy parronts.

Note. You can ask a CITES official or a customs officer or a vet or an airline or the forestry department or the health authority what it takes to travel internationally with a parrot and they will all tell you what they know, but please believe me, none of them know everything involved. The only way to find out, is to contact every one of them, to find out what each of them requires.