Parrots can not keep up with flying players.

Like cats and wolves, Parrots can't keep proper track of fast moving players, however unlike those land-bound creatures parrots can not do something that should be expected- fly along with a player who is gliding with an Elytra. Unless a player lands every few chunks the parrot will be lost and they will need to back track to hopefully find them.

A more appropriate behaviour would be for the parrot to fly alongside the gliding player, including firework-assisted flight, so that they won't get lost.