How to keep pets from teleporting

I have a small parrot room, with a glass top. The point is for the parrots to act normal, and for players to view them from the top. The only way for the stubborn parrots to cooperate was to make them my pets, and put them in there. BUT, (because there always are problems) the parrots won't stop teleporting to me when I walk away. I could make them sit, but that would defeat the purpose of a free-range type thing... Does anyone know of a command or something that will stop the teleporting without mods?

Solution 1:

You can put a leash on untamed parrots and transport them into the cage that way. If you're really careful, you can even fly with an elytra while having them dangling behind you.

Xisuma did this in a Hermitcraft episode, but that's hard to find, so I'll link you a flying sheep video instead: