Kill command has no effect on parrots perched on player's shoulders

Entering the following command to kill all parrot entities will not kill any parrots that are perched on the player's shoulders:

/kill @e[type=minecraft:parrot]

If the only parrots in the game are those currently on the player's shoulders, an error is displayed:

No entity was found

Killing all non-player entities has no effect on them either:

/kill @e[type!=player]

Killing all entities detaches the parrots from the player's shoulder as the player dies, but leaves the parrots alive:

/kill @e