Minecraft wiki entry for Parrot, associated text: does not land on a high-enough surface (1⁄2 block up or higher)
drops off a ledge of higher than 3⁄4 of a block
takes damage
submerges the player's feet into the water of any height
starts drowning
sleeps on a bed
submerges the player's head in lava (the parrot dismounts and burns even if the player has Fire Resistance)