Found when trying to make tamed parrots sit in a 'birdbath' which included a waterlogged slab surrounded by signs to keep the water from flowing. I noticed that the parrots were never in the birdbaths whenever I left and came back. After watching them more closely for a time, I spotted that they were moving on their own even though they were sitting. They even remain in the sitting animation as they move or fly away.

I discovered this on a multiplayer server (using Vanilla), so I tested it in singleplayer to find that it does occur in local worlds as well. I also tested sitting parrots on other blocks like dry slabs and walls, and found that only those in contact with water seem to 'forget' that they're sitting. Most often, I found that unloading and reloading the chunk is the most reliable way of replicating this glitch, though I have seen them start moving when I've had the chunk loaded for a while.

Steps to Reproduce:

1. Set up a waterlogged slab so that the water doesn't flow beyond the slab (can be done with signs, iron bars, or even a debug stick)
2. Push a sitting tamed parrot onto the slab (I do recommend pushing it when it's already sitting - if you try to dismount it from your shoulder, it will usually just fly away)
3. Move away from the area so that the chunk and parrot(s) are unloaded (I've also found that quitting then loading back into the world makes this occur more reliably than just leaving and coming back)
4. Return to the chunk with the sitting parrot, and it will most likely start to fly away - if not, repeat the above steps and/or try quitting and loading back into the world

Note: In the attached video, I used Optifine zoom to show the parrots' lack of flying animation up-close, but I did test in Vanilla to make sure it wasn't Optifine-related.