That Guy,
I'm not certain that they allow overnight parking, much less how much it might cost, but you could consider New Haven. They do have a big parking facility right there, and it would be a straight shot right down I-91 to the station. You can call the New Haven Parking Authority to inquire about costs and overnights at 203-946-8936.
From New Haven, you have 2 options, buy tickets for the cheaper Metro North trains and then transfer from Grand Central to Penn Station in NY. Or buy Amtrak tickets and go straight to Penn. Amtrak is also faster, since it makes fewer stops. Be sure to allow enough time for delays, peferably at least 2 hours between connecting trains.
Being that you're in a sleeper, once you get to NY Penn, you can go to the Club Acela to wait for your next train. As a sleeping class passenger you are entitled to utilize Amtrak's first class lounges in NY. These lounges are a nice quiet refuge in the often-bustling train station. They have nice plush chairs, conference rooms, computer terminals, newspapers, TV's, luggage storage, free soft drinks, juices, coffee, tea, and clean safe bathrooms. They also allow you to board your train before the rest of the masses.
The lounge in NY can be found to the far left of the NJ Transit ticket windows as you face them, when standing in the Amtrak concourse which is on the 8th Avenue side of the station. Walk past the West Gate for tracks 7 & 8 on the left side of the escalator and you will see a sign on the corner of the wall. There is a doorbell to the left of two gold doors. Ring the bell and wait for the click to open the door. Note: In NY there is no pre-boarding ahead of the regular passengers, but they usually do announce your train in the lounge before they announce it in the main station. This means if you don’t dilly-dally, you can generally be near the head of the line at the escalator.
If New Haven isn't viable, try to find a parking lot west of 9th Avenue for cheaper rates in NY. But one problem that I see is that not all lots are staffed 24/7. So even if they let you overnight, there may not be anyone to keep an eye on your car. That could be very dangerous.
Another alternative migh be to drive to Albany, where overnight parking is also permitted and board an Amtrak train from there. That's about a 2-1/2 hour ride on the train from there.