Tip for anyone that needs to park at JFK and wants to save some money-
Right next to the Howard Beach airtrain/A station there is free on street parking. I am pretty sure there in no alternate side there, too.
You will need to pay the fee for the airtrain to get to the terminals and it is about a 5/7 min ride. The fee is $5 for those with a metrocard or $6 if you need a new one.
I have done this in the past but only for an overnight trip.
Note- the neighborhood may not be the safest neighborhood to leave your car.