Is there any sign at the entrance do indicate that there is a height restriction?
No, there were no signs at the entrance to indicate a maximum height restriction in the car park (hence choosing it).
Is the physical height restriction only at the exit?
Yes. They physical height restriction was only in place at the exit.
Is your vehicle unusually high?
The vehicle is a High Roof Long Wheelbase Ford Transit. So, it is no higher than a minibus or vehicle of a similar height.
You can see an example here.
Was there anything at the entrance to the car park to indicate the commercial vehicles were not allowed?
No. There were also other commercial vehicles parked in the car park, near the entrance.
Are you saying that you passed under the height restriction barrier at the exit but still damage the vehicle roof as the car park roof was actually lower than height restriction barrier
No. There was no roof to the car park. It was an open topped ground level car park. (Not multi-storey and not underground)
I damaged the top part of the front slope of the van- so there was no way to physically fit through the barrier (I found this out after colliding with it). I had to turn the van around and drive out through the entrance after finding the van wouldn't fit.