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