The bug

Feeding a parrot a cookie does not count as killing.

Note: This is probably the reason why you cannot kill an Invulnerable parrot this way in Creative mode (see MC-117593).

How to reproduce

1. Create a scoreboard objective tracking how often you killed a parrot

   /scoreboard objectives add killedParrots stat.killEntity.Parrot
   

2. Have the objective be displayed on the sidebar

   /scoreboard objectives setdisplay sidebar killedParrots
   

3. Spawn a parrot and kill it by punching it to make sure the objective works
4. Spawn another parrot and feed it a cookie