A Tamed Parrot will attempt to get on your shoulder when you are riding a Tamed Donkey, when you then get off the Donkey, the Parrot remains sitting on the Donkey and you are no longer able to get back on the Donkey, you can access the Donkeys inventory by right clicking and remove the Saddle, but the Parrot still remains on the back of the Donkey.

How to recreate, Tame a Parrot, Tame a Donkey, ride the Donkey with a Saddle attached, wait for the Parrot to fly towards you and attempt to settle on your shoulder, get off the Donkey, see the Parrot on the Donkey, attempt to get back on the Donkey.

Hopefully that's clear enough, and worded correctly!