Can you tow a trailer with my truck on parkway?

You need to be more specific. In the Northeast USA and especially New York It's a safe bet to say NO YOU CAN'T. In fact in NY, Parkways are restricted to Cars, Motorcycles, SUV's and Light Weight Pick up trucks only.

Commercial Vehicles such as: Larger (over 5000 gvw) Pick up trucks, Campers, anything towing a trailer, semi's, dump trucks, box trucks like U-Haul Trucks, Pensky, Ryder rental trucks are prohibited. Few of the parkways have bridges that meet the 13' height rule. In other Words, most box trucks, semi's and trailers will NOT make it under the bridges.

When in doubt, contact the local State Police in the area you are traveling in and they can tell you exactly.

There maybe an exception on the Belt Parkway, because of the people that live in Sheepshead Bay need to get their boats out. I'm trying to find out an offcial Word though