NB this is a free parking space - it’s necessary to provide reg no. at the bar. If pub closed for refurb, how can driver inform pub?
That's not quite true, as the signage you have shown us states, it is free parking
for customers of the pub. To flip your question round, how can the driver be a customer if the pub is closed?
It would seem that the driver was trespassing, but that does not necessarily mean he has entered into a contract with Park Direct.
There is a potential argument to run that the signage is forbidding, in that it doesn't make an offer to non pub customers. This argument would need to be made in court, neither the operator not the IAS would consider it.