I'm not 100% sure this is a bug or an intended effect which many of us players would possibly like changed.
When the user jumps, the parrot gets off their shoulders, parrots can easily hold onto a real person's shoulders when they jump or run, making this an unequal real life equivalency.
Many of us who play with out friends would like to easily take our parrots on adventures without fear of accidentally loosing them into lava in a cave or from any other possible ways to lose them from dismounting,
From yours - Yumiko