Like tamed wolves and cats, a tamed parrot follows the player unless told to sit, and may teleport if there is a sufficient distance between them and the player. ... Parrot teleportation is completely silent, which is intentional.

Can parrots teleport?

To transport parrots you need to put them on your shoulder. However jumping removes them from your shoulder so you'd need level ground between that spot and your boat. One thing to remember: the edge of your view distance is also the edge of loaded land. The land beyond it is saved to disk and is frozen in time.

Why are my parrot not teleport in Minecraft?

Parrots, or any tamable animal. Will not teleport to you until they are in a loaded chunk. If the chunk is unloaded the entities are also unloaded, therefore they practically don't exist. Try getting the animal inside the boat with you.