Can I park at JFK and use AirTrain if I am not flying?

Nothing in the Port Authority's JFK parking rates and information page says anything about parking being reserved for passengers; certainly, I have never seen or heard of anyone being asked to present a boarding pass or other proof that you are there to do business at the airport.

But if you're coming from Virginia to go to a show in Manhattan, parking at JFK is an absolutely atrocious idea. JFK is in Queens, so you're overshooting, crossing two bodies of water for the privilege with all the tolls and traffic that entails, and then backtracking an hour via AirTrain, LIRR, and subway.

For Broadway shows, I've parked before at the Port Authority Bus Terminal, which is walking distance from the theatre district. At $35 for 12 hours, it's reasonable by Manhattan standards. But presumably you wanted to avoid the multiple hassles of driving into Manhattan in the first place, in which case you should seek parking on the New Jersey side, perhaps at an NJ Transit station on a line with service to New York Penn Station. Rates will be cheaper than for EWR parking, and you avoid having to pay AirTrain fare.

I don't know how many people are in your party are coming up from Virginia, but if it's just one or two, you might also consider taking one of the numerous DC-NYC bus services, which can get you there for under $50 round trip— less than tolls, gas, and parking would be for a small party. Amtrak is faster, more reliable, and more comfortable, but not price-competitive with the buses.