Thank you for your comprehensive answer. I would like some clarifications.
1) I've read that it is worth asking for them to prove a contract exists between them and the owner of the site. They are likely to refuse this and not take the matter further. Did a contract actually exist between them and me?
2) Is proving loss worth considering before it went to court? It's not clear who has lost on their side. The owner of the site received payment. The reinforcement firm has lost the cost of 2 letters.
3) The fact they accepted my argument on the first 'offence' might be considered a precedent for not taking the matter further?
4) I would have thought that (2) above was the best defence using this Dept. of Transport extract below, especially as there were no unpaid charges. Do you agree?
Charges for breaking a parking contract must be reasonable and a genuine preestimate of loss. This means charges must compensate the landholder only for the loss they are likely to suffer because the parking contract has been broken. For example, to cover the unpaid charges and the administrative costs associated with issuing the ticket to recover the charges. Charges may not be set at higher levels than necessary to recover business losses and the intention should not be to penalise the driver.
It could also be argued that if they are concerned soley that a parking charge has been paid, they already have that information on the machine that issues tickets (which includes the car registration), and that seeking a £60 payment is a charge at a higher level than necessary to 'recover business losses'.