The Tame:1 tag doesn't work on parrots

When you summon a parrot with:
/summon parrot ~ ~ ~

{Tame:1}

It summons just a normal parrot, not a tamed one.

It doesn't even work with entitydata commands:
/entitydata @e[type=parrot]

{Tame:1}

Meanwhile, it does work for other entities, for example - llamas:
/summon llama ~ ~ ~

{Tame:1}