It's 5pm and your date is at the door. As you run out the door, your parents madly snapping pictures, your mom asks, "So, what are you doing after the prom?" Great... Ideally you would have approached this subject earlier, but maybe you were both too busy and forgot, and now you're in a crappy situation.
If your date or friend is listening, ask them for a minute alone with your parents. It's usually not a good idea to ask your parents for permission for something in front of the people you want to do it with. It puts everyone in an unfair position, and that's not the way to get what you want.
Privately, tell your parents what your plans were. Then follow steps from Plan A. Respect their response and try to compromise if you don't see eye to eye.