I'm in NY too and my 9 yr. old TT is paid off. My car insurance covers the trailer when towing but only for liability.
I do have a separate policy that covers damage to my trailer if it should get hit or damaged in an accident. It also covers in case a tree branch snaps off and damages my parked trailer.
I do not have ins. on my tilt bed and dump trailers, only the liability when towing.
The short answer for you is NO, you have liability ins when towing but if you back your trailer into a tree, car or anything and damage the trailer you will have to repair the trailer out of your pocket. Same while parked. If trailer gets damaged, no coverage for repair.