OK thanks! will they stay while I am flying with elytra? and what happens if I die?
Will they stay when flying? No. They usually stay on your shoulder if you jump up one block. But when you fall/land they come off. They then will fly around and try to follow you. But if you fly with an elytra. You'll be going to fast for them to keep up.
If you die? I'm not sure. I imagine they'd stay where you died. But if you do /back you can go there and pick them up again. There is an instance where they go back to spawn. I can't remember when this happens though. So if they disappear, go check the spawn location in the windmill. I remember in survival 1 there's an army of parrots at spawn.