Was the vehicle parked in an "Accessible Bay" for blue badge holders? This is crucial information that we need.
You do have grounds for an appeal. You should also not identify who the driver is. There is no legal obligation on the Keeper (presumably you) to identify the driver. There is a technical flaw in the Notice to Keeper (NtK) that means that PE cannot transfer liability for the charge from the driver (unknown to them) to you, the Keeper (known to them).
Whilst PE will argue that the NtK is compliant with PoFA 2012, the law that allows them to hold the Keeper liable if they don't know the identity of the driver, the technical flaw is that there is no "invitation" for the Keeper to pay the charge. PoFA 9(2)(e)(i) requires that the NtK "invites" the Keeper to pay the charge. There is no "invitation" not any synonym of the word in that NtK. To be able to transfer liability from the unknown driver to the known Keeper, the NtK MUST fully comply with ALL the requirements of PoFA. Partial or even substantial compliance is not sufficient.
The other points of appeal will be breaches of their own AOS (the BPA) Code of Practice (CoP). If the vehicle was parked correctly and had paid for the parking, they have to allow a "grace period" during which no parking charge can be made.
Section 16.1 of the BPA CoP states:
Where parking is being provided as a service, you need to have regard to the obligations placed upon you by the Equality Act 2010 and the Services, Public functions and Associations: Statutory Code of Practice, issued by the Equality and Human Rights (EHRC), in particular to make reasonable adaptations to accommodate disabled people. Reference should also be given to the car park accessibility section in Inclusive Mobility, guidance published on GOV.UK by the Department for Transport on the creation and maintenance of an accessible and inclusive built environment and public realm, an essential document for those seeking to produce an inclusive environment and meet the requirements of the Act, including the public sector Equality Duty, and other legislation.
Many parking operators and landowners choose to recognise the Blue Badge scheme and provide designated provision with specific bays which allow more space for opening vehicle doors, getting in and out of the vehicle, accessing a wheelchair etc. This might be appropriate for recognising the needs of people with limited physical mobility, but adaptations are not purely physical - people with other disabilities might reasonably need longer consideration period and grace periods, more time to access payment machinery, and other ways to pay where payment is required. Recognition of these obligations is important in the consideration of appeals.
In any initial appeal, it will be necessary to highlight this rule and their failure to comply with it. You will be required to show a copy of the blue badge.
The latest Joint CoP shows that all private car parks open to the public must provide a minimum 10 minute grace period if all other requirements of the terms have been complied with.
Please let us know the answer to my first question about whether the car was parked in an accessible bay for blue badge holders.