Wandering view,
You can take the Garden State Parkway. However, there are one or two arched overpasses which are low on the rightmost lanes. They are well marked, so just shift to the center lanes and you're good. I have travelled that route in our motorhome several times towing our Odyssey. No problems.
However, tolls will KILL you on 95, North and South. Probably at least $100 one way depending upon your actual route. (Our gasser Class A towing the Odyssey would be $60 on the GW Bridge Northbound in NYC! And that's Off-Peak with the EZ-Pass Discount!!! It's considered a 4 axle vehicle and gets classified with the trucks.
GW Bridge Toll Rates.) The route I show below avoids all of the tolls except crossing the Hudson River, and those are only in one direction, heading toward New England. I'd recommend getting an E-Z Pass for your rig if you don't already have one. Especially if you'll be taking 95. Several of the toll plazas allow you to stay at speed so you don't even have to slow down for the toll booths.
If you're not going to be travelling in the winter (snow season) or if you are and the weather permits, I'd recommend taking 84W to 81S to 77S to 26E to 95S. It looks like a long way around, but you'll use less gas than you'll spend in tolls taking 95. Plus it's a much more pleasurable trip. 95 is like a racetrack until you get below Richmond.
We're scheduled to head down between Christmas and New Years and I'll be watching the weather closely to see which route is based based upon the weather conditions. If weather isn't an issue, then I'll be taking the inland route.
Have a great trip!
~Rick