The DVLA will hand out your personal details under their "reasonable cause" provisions but timings are very relevant. The Parking company must either send a parking charge by mail to the Keeper to arrive within 14 days. Or if it was a paper ticket on the car, they cannot ask for Keeper details earlier than the 29th day nor later than the 56th day. Often the Parking Contractor fails to do this.
So you should check the Parking company has held to these timings by getting the DVLA to tell you when the Parking company requested your details and when they were given them.
To get this information you send a Vehicle Subject Request. There is no charge for your own details. Send by normal post and remember to sign it as they may want to check the signature - Data Protection Act. You can also try emailing SubjectAccess.Requests@dvla.gsi.gov.uk. See this link
DVLA Vehicle Record Enquiries section
Re: VRM AB12 XYZ
As the Registered Keeper of the above VRM could you advise who has accessed my personal details with regards to this marque, how often and when did the DVLA send the keeper details out. Please advise the information with regards to events between xx/xx/xx/ and xx/xx/xx.
I understand there is no charge for this information and look forward to your speedy reply.
Mr Registered Keeper