The bug

Parrots dismount as soon as your y coordinate decreases. This creates inconsistent situations for blocks you can walk up without having to jump, walking up does not dismount parrots while walking down does. Since parrots dismount even for the slightest y coordinate decrease they dismount when walking down carpets or lilypads which are less than 0.1 blocks tall.

How to reproduce

1. Spawn and tame a parrot
2. Let the parrot sit on your shoulder
3. Walk up carpet
4. Walk down from the carpet
   → The parrot will dismount, even though you didn't jump