In a dogfight, can a pilot shoot down a parachute?
You question is a question of rules of engagement. Shooting down a downed pilot in a dogfight is not only not OK, but also illegal. 

According to Article 42 of Geneva Conventions, 


  No person parachuting from an aircraft in distress shall be made the object of attack during his descent.


Of course, war is war and this kind of attacks have been practiced by pilots of almost all air forces, though it should be noted that in most cases, this practice was specifically prohibited by their high commands.