If you leash a tamed parrot to a fence, then put the parrot on your shoulder, then move away (presumed 10 blocks, since that is the length a lead will go to without breaking) then jump, the parrot will leave the shoulder and the lead (item) will drop on the floor by the player, the lead (visually) still appears on the fence. This only appears to be a graphical glitch, as no other leads drop.