I'm actually considering a lot of this, and thinking about how to maneuver through obstacles. The AKC is going to be my "model" for the most part. Someone suggested a band code registry so others couldn't just order bands of the same codes. Each breeder would need their own.
As for initial registry, it would have to coincide with the first "national" show. There, birds would have to be registered in person so we could see and compare traits. I take my scarlet for example. He's clearly a scarlet. But I am thinking he may have a Catalina or another hybrid mix somewhere way back, because he has blue and green on the back of his neck in small patches. It's not enough to be obvious, but it's there.
So no, he probably isn't pure, but he is a "scarlet" macaw. So he would be eligible for registration as a scarlet macaw, and regular confirmation shows would encourage someone to breed those extra colors out.
Registered animals get a certificate. When they breed, broods are registered as a whole by the breeder, using parents' registration information. Individual certificates are issued for each offspring for the NEW owner to register after adoption. That is how the AKC does their registration process. But for parrots, registration would also need to be held at regular "national" shows for several years to allow the entire parrot trade to get on board. One or two "judges" could also be willing to travel to large aviaries (for a fee) to help with their massive registrations.
There's a lot more to think about and work on, but I feel motivated right now. Keep throwing questions and ideas!