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
Spawn and tame a parrot
Let the parrot sit on your shoulder
Walk up carpet
Walk down from the carpet
→ The parrot will dismount, even though you didn't jump
This was the "fix" for MC-114889.