How long is a piece of string?
I have never come across this company previously. They are not Approved Operator Scheme (AOS) members of either the BPA or the IPC. If a parking company is not an AOS member of either the British Parking Association (BPA) or the International Parking Community (IPC), they cannot lawfully obtain registered keeper data from the DVLA for private parking enforcement purposes.
The DVLA provides vehicle keeper data under Regulation 27 of the Road Vehicles (Registration and Licensing) Regulations 2002. This allows access to data only when there is a reasonable cause. To ensure compliance with this regulation, the DVLA requires parking companies to be members of an AOS operated by either the BPA or the IPC which are Approved Trade Associations (ATA). Membership demonstrates that the company adheres to a recognised code of practice and provides a mechanism for resolving disputes.
Membership in an AOS is considered evidence of reasonable cause to request vehicle keeper data. The AOS ensures that the parking company meets minimum operational and ethical standards, including providing independent appeals and clear signage (no sniggering from the back of the class please!). Without AOS membership, the parking company cannot demonstrate compliance with these standards, and the DVLA will not release data.
If a parking company such as ParkJockey obtains data without AOS membership, they may have done so through fraudulent means, such as falsely claiming they have reasonable cause. They may try and use a third-party contractor (e.g., an AOS member requesting data on their behalf). If this happens, the third party will be violating the DVLA's terms.
Either method would breach data protection laws under the UK GDPR and the Data Protection Act 2018, as well as DVLA rules.
So, it is unlikely that you will receive a Parking Charge Notice (PCN) by way of a postal Notice to Keeper (NtK) from this company and if you do, they will have likely breached your GDPR which would open them up to a compensation claim from you for having done so. Trying to pre-empt a PCN was probably not the correct thing to do under the circumstances but a payment was made and any allegation of a breach of terms will depend on what exactly the terms on the signs said.
So, unless you receive a PCN, get on with your life. If you do, come back and show us.