The simple answer is yes. If you contact a parking company, you (the Registered Keeper) can ask for anything they have relating to that registration (vehicle registration mark or VRM), what they did with it and who they sent it to.
The ICO explantation from a complaint about a parking company.
I have explained to [the parking company] that, under the DPA, personal data is defined as ‘data which relate to a living individual who can be identified
(a) from those data, or
(b) from those data and other information which is in the possession of, or is likely to come into the possession of, the data controller’ [..at the parking company].
An individual is 'identified' if you have distinguished that individual from other members of a group. In most cases an individual’s name together with some other information will be sufficient to identify them. A name is the most common means of identifying someone. However, whether any potential identifier actually identifies an individual depends on the context.
Where a vehicle registration mark (VRM) is collected (often as part of an automatic number plate recognition (ANPR) system), the ultimate purpose of which is to identify and take some action against a living individual (such as to serve them with a parking fine) the VRM will be personal data at the point of collection. This is because the data controller is likely to have or come into possession of further information which will allow them to identify either the driver or registered keeper of the vehicle, or both.
Therefore, it can be seen that the vehicle registration number constitutes the personal data of Mrs [xxxx], regardless of the fact that her name was not disclosed in this case.