Now that I have your attention...
Suppose it took two hits to kill a Parrot. It'd take 0 damage from the first attack to hit it, as an in-built dodge ability.
I suggest this because the Parrot is easier to shut down than any other weapon, and because of that, often fails to distract or damage opponents (me) for a meaningful length of time.
One can turn and aim in awkward directions to influence the Parrot's trajectory, but it often flies in an only slightly-curved line towards its target, making it easy to hit en-route. This is where I feel the extra life would have the most impact.
With more time to close in, a Parrot could reach and begin flying around its target's head (where its flight pattern truly makes it bothersome to swat) from even a moderate distance.
As much as I would despise such an outcome, it would also lend validity to the strategy learned fastest by new Captains. Bird, gun, reload.