Yes this is legal provided there are clear signs in the car park advising motorists of the terms and conditions under which they use the car park.
Motorists should expect at a minimum to see large signs throughout the car park and clearly visible to inbound motorists advising that
The signs should provide the name and contact telephone numbers for the car park operator/manager and/or the parking enforcement contractor.
The signs in the car parks clearly identify the management company for the particular car park location. You should gather all of the evidence to support your appeal including copies of permits, photographs of signs etc and write to them setting out your case. Set out your case in clear concise English. Avoid any temptation to vent your unhappiness . Focus on clear logical reasons as grounds of appeal e.g. P&D machine failed, tickets were mis-printed, machine refused to accept coins, signs were obscured/damaged/illegible, warden failed to see permit on dashboard etc.
If you write directly to the property owner they will refer the appeal to the car park management company so you are best advised to start with them. Keep copies of all correspondence.