Parrots can be picked up while leashed to a fence

Leash a tamed parrot using a lead and attach it to a fence post. Mount the parrot onto your shoulder. Notice how the lead visually vanishes.

After jumping to dismount the parrot, the lead will attempt to reattach to the fence. If the parrot is too far away, the lead will snap. Otherwise, it will reconnect and yank the parrot back to its original place.