New Haven has pay parking lots and parking meters - I don't think you'll find free parking near campus.

Shuttle if possible

Traffic can be/is likely to be terrible, and unless you are a former New Yorker, you won't know the workarounds. Leave the driving to a pro.

There is essentially no free parking on the street in safe urban areas in the northeast. Meters and/or neighborhood parking stickers are very common.

There are various ways you can make the trip from NYC but I don't know them well enough to know how they would fit your needs.

It would be useful to others to know when you arrive at JFK, when you need to be in NH, and when you need to be back at JFK or go on to your next destination.

This is not a drive you want to do if you don;t have to - esp if you are anywhere near rush hour.

If you provide more details people may be able to give better advice,

If you don't need a car while in New Haven, then take the shuttle. It's a lot of hassle and expense to find a pay for parking there.

If it weren't such a hassle I'd say take the train, but it's difficult to get into Manhattan, change train stations, and get back to New Haven in a reasonable amount of time, though trains are fairly frequent. (You can take Amtrak without changing train stations, but it's pretty expensive, though it might be an option if you want to avoid driving or taking the roads entirely.)

Though I still don't agree with the doom and gloom crowd. Driving on I-95 isn't pleasant, but it's pretty easy and straightforward to get to New Haven. But if you do decide to drive, make sure you have a GPS or at least plan out your route on Googlemaps before hand. First-time drivers in NYC find it difficult to get into and out of the city. We have pretty bad signage.