Using a firework rocket (whether by hand or shot from a crossbow) damages and can kill entities due to the firework explosion.

The bug is that damaging or killing a mob in this way does not get counted down for advancements which you get by damaging or killing them, while it does count for things like bows and arrows, splash and lingering potions, and even explosions from TNT.

Steps to reproduce:

1) Summon a zombie or other mob that counts towards the "Monster Hunter" advancement.

2) Craft a firework rocket, and use the firework rocket (either by hand on the ground or by shooting out of a crossbow) to kill the zombie.

3) Observe you do not get the "Monster Hunter" advancement.