^{Update by EVGENSYPERPRO}
Summary:
https://youtu.be/DA2pi7Kx9gQ?t=78 as seen in this video you used to be able to fly using an elytra with parrots. They should stay on the shoulder when only flying up. In some snapshots parrots dismounted during upward movement but it was changed that they dismount the player’s should when falling.

Steps to reproduce:

1. Summon/find a parrot.
2. Tame a parrot.
3. Sit the parrot on your shoulder.
4. Put on the elytra.
5. Fly up on the elytra.

Observed results:
When flying up on an elytra with a parrot, the parrot dismounts.

Expected results:
When flying up with a parrot, the parrot must not dismount. The parrot should dismount only when flying down.

Original Description:
I really wanted to report this but Parrots are not on the players' shoulders when flying the Elytra.
It was a problem from since like 1.12 Pre-2.