The info you need is basically a combination of all these answers, I assume all you have at the moment is a ticket which was left on the car, with the wrong reg number (all be it close).
As said they won't be able to trace you as they don't have your actual number and in that case I'd ignore it.
up4it and north star are correct in that a private company cannot issue a fine, all they can do is enter a contract with you and all they can claim is a direct loss as a result of the breech in contract. As said the most you would be liable for would be the loss in parking fee they have incurred due to you parking in the bay as north star points out.
This site explains this in much more detail and I would advise you read it...
http://www.consumeractiongroup.co.uk/forum/par king-traffic-offences/65341-private-parking-co mpanies-charges.html
As New Judge says they can chase the sum as a civil debt however if they went down that route all you do is contact them at the point that start the proceedings and use the information in the above link.
As a small side anecdote, a friend who worked for one of these companies told me they hardly ever get to the stage of getting DVLA details as this is an expensive process, almost never used for individual cases at his particular company, he said they made more than enough money from people who paid up without a fight.
In summary , ignore it until such time as they send you a letter chasing the debt (which I think we all agree won't happen), at that point state your case that the fine has no legal standing using the link about and that you will pay them the appropriate level of compensation (�1.00).