JFK has long term parking on site, I never use it but in this case, with a dog I probably would recomment it.

Details over at www.panynj.gov

When you get back, leave the dog in the carrier and head over to the parking lot.

I usually offiste lots, airparkparking.com there, I would give them a call and see if a dog in a carrier is OK for them before you park your car there.

You have cleared all the issues with bringing a dog into the country, right? Make sure you check that out, it's not as simple as just showing up with it.

They have lost dogs at JFK, make sure the carrier is marked, and make sure the dog has a collar on with full ID on it.