Yes, Arrods is quite right. Assuming he drove to the private land on public roads he would need insurance to do that and his policy would almost certainly cover him for damage to Third Party property, even if the damage occurred on private land.
However, the matter of insurance is not really the issue. Whether he needed insurance or not (and whether he had any or not) he is still liable to make restitution for any damage he causes. Having insurance simply covers him for that eventuality but having none does not make his responsibility void.
A sign that OG suggests might exist (but it seems does not anyway) would not absolve individuals from liability. Those signs are usually to protect the site owner or landowner from responsibility.
You question whether either company has a “duty of care” in the matter. A duty of care only extends to a person’s health, safety and wellbeing. You’ll find it a bit of a stretch to get it to encompass damage to vehicles.
This is quite straightforward. Another vehicle collided with yours whilst it was parked. The driver of that other vehicle is liable to pay for the damage to yours. How he does so (whether out of his pocket, via an insurance policy or via his employer) is for him to determine.
One thing you should do is inform your own insurers of the incident. Most policies contain a condition that you should inform them when involved in any incident of this type. Apart from that, if you are unsuccessful in getting the other driver to shell out you may want to claim on your own policy. If you don’t tell your insurers promptly they may refuse cover. Whilst your digging out your policy you might have a look to see if you have legal cover. If so, that may be a way to start the ball rolling.