I would have thought that as you paid for your parking up front that by them contacting the DVLA claiming you hadn't paid for parking was when the breach of GDPR took place.If they continue to contact you re the ticket, that would be a further breach of the GDPR.
To illustrate your position there is the case some time ago regarding Baroness Walmsley and TFL. She paid the ticket and entered the wrong registration and ended up in the High Court where it was established that it is not the purpose of the scheme to penalise those who make a genuine error. A fact that should be well known to the parking companies since some of them will have lost in Court over the same situation. So as I said, the GDPR breach was theirs when they asked the DVLA for your data.I have included the relevant section below from her case.
. 43. This interpretation of regulation 16 has the effect of reconciling the provisions of the Scheme as a whole with its purpose, which is to ensure that charges are paid for cars that enter the Zone and that those who fail to pay are penalised. It is not a purpose of the Scheme to penalise those who make a genuine error as to their vehicle's registration number. As has been seen, many people do make such errors and are relieved of penalty. It is and must always have been obvious to Transport for London that there were bound to be many people who would mis-state the registration numbers of their vehicles. For example, it is obviously easy to confuse the letter I with the numeral 1, and the letter O with the number zero, quite apart from the room for mistyping or simple mistake as occurred in this case. 4
Don't touch them owe me £500 since January 2019 make excuse after excuse. Seem they always have software problems sending money out. Keep saying they will call back or email nothing been chasing it now for 6 mths the phone staff always have the same banter we will chase it up and get back to you then nothing!