Parrots with the invisibility effect have their model rendered anyways when they are on your shoulder.
Side note I tried to get the data of the parrot whilst it was on my shoulder and the command didn't find an entity, so I think what happens is that when it goes on your shoulder the entity is temporarily removed from the game and that's why it happens, supporting this theory, other potion effects like poison don't take effect while it's on your shoulder as well.
RedFoxLightning
Thank you for your report!
We're tracking this issue in MC-115111, so this ticket is being resolved and linked as a duplicate.
That ticket has already been resolved as working as intended, which means this is not considered a bug and won't be fixed. Please do not leave a comment on the linked ticket.
If you haven't already, you might like to make use of the search feature to see if the issue has already been mentioned.
Quick Links:
📓 Bug Tracker Guidelines – 💬 Community Support – 📧 Mojang Support (Technical Issues) – 📧 Microsoft Support (Account Issues)
📓 Project Summary – ✍️ Feedback and Suggestions – 📖 Game Wiki