You don't have to pay to access the AirTrain platform from the long-term-parking area. You pay only when crossing between the AirTrain area and the "rest of the world" (which includes the subway).

Which means if you somehow needed to access the long-term-parking area from the street at that point, or vice versa, you'd need to pay for the AirTrain even though you wouldn't be using it.

Seth