A lot of airport hotels have "park and fly" rates where you get a number of days of free parking if you stay a night. If you google "JFK park and fly" (without quotation marks) you'll find a bunch of websites listing these hotels. The Ramada (which is supposed to be a terrible hotel) offers 10 days of parking. And according to this site, the Days Inn offers up to 14 days. I'd try calling the hotels directly to check on these rates. And note that the extra parking days may not be included if you don't book a rate that specifically includes it. Also note that not all of these hotels have 24 hours airport shuttles (although some do), so you may want to factor in an extra $10 or so each way for a car service or taxi.