These signs have recently gone up on the land behind the shops on Shirley Road (accessed from beside Jeffries Hardware). Existing car parking immediately behind Jeffries isn't affected, but all the space beyond that is.
One thing that this government got right was to ban private wheel clamping on private land - that was simply extortion. However, it hasn't stopped parking companies issuing parking tickets - often termed as "Parking Charge Notices" so they look like council-issued Penalty Charge Notices, even though they aren't.
When you enter a privately run car park like this, the operator considers that you are entering into a contract and if you break that contract, then they will invoice you. They are not "fines" in the same way that a council or police-issued ticket is. The company has to go to court to obtain judgement against you - unlike the police or the council.
The excellent Moneysaving Expert has a guide on how to appeal dodgy fines. As GBPS are a member of the British Parking Association, there is an independent appeal process and a code of practice.
Always take care in where you park and unless you can see clear signs allowing you to park there, don't take the risk. These firms are after your cash, pure and simple.