1. Go to creative mode
2. Tame a Parrot
3. fly up into the air
4. Right click on your parrot
5. Go to 3rd person mode
6. Fly up or down
and you should not see your parrot on your shoulder
unless you get on the ground and jump to get rid of them then the are visible
Tested with multiple parrots, only in Singleplayer(Opened on lan to use cheats), with one person
Linked issues
Comments

Thank you for your report!
However, this issue has been closed as a Duplicate of MC-114889.
It has been linked to this report. If you have additional information, please add it to the duplicated report.
Please search before reporting, as it's likely that one exists already.
Quick Links:
📓 Issue Guidelines – 💬 Community Support – 📧 Customer Support – 📖 Game Wiki
You're right, I just tried this, it seems it's despawning too. Edit: It comes back after a few minutes.