For those of you reading this thread for extra tips on getting this assignment, even though the OP has already gotten it, here ya go:
The airdropped IFV is either an LAV or BMP (depending on the faction) that you can spawn in from the spawnscreen menu when your team controls the C base (or whatever has the plane icon next to it) on Conquest. It'll read something like "Dropship IFV." If it isn't there, either your dropship isn't in the sky, is out of zone, or someone has already taken the dropship IFV and it must be destroyed in order for it to respawn.
Upon spawning in the IFV, you enter an animation where you parachute out of the dropship in an IFV, but the turret is usable. Once you land, it is a controllable IFV.
The best way to destroy an enemy IFV when already in the dropship IFV is to simply play the objective, but also taking care to stay alive. Have APFSDS-T shells equipped, that's the best way to destroy an enemy IFV. Press Y/Triangle to switch between that and the normal shells. Fire both at the enemy IFV alternately and try to keep moving until it's destroyed. Also, a good way to defend your IFV is to have IR Smoke, which keeps you save from all guided weaponry upon pressing RB/R2 (double-check PS3).