Can the yard owner not find who was rented the spot next to you? The lot where I park doesn't have designated spots but the owner has records of all TTs and owners stored there. Should be able to find matching damage on one of those trailers if it returns to the lot or track down ones no longer there.
Either way, damage that occurs on private property will likely have to be covered by your own insurance company regardless of who is at fault. My wife got backed into while parked in a grocery store parking lot. The other person was at fault but it was still my insurance that had to cover because it happened on private property.