So far I have not been caught by private parking enforcers, but I have some ideas if I ever am. For example, if I'm in breach of contract with anyone, it's with the owner of the supermarket or whatever, not with the parking company who manages the car park on its behalf. So, ask for a signed statement from a director of the supermarket company that XYZ parking is acting as its agent and is empowered to collect money on its behalf. Alternatively, if they take me to court, insist on the supermarket company sending a representative as a witness to the contractual arrangement.

If I did send a cheque, to avoid money laundering I would make it payable to the supermarket, not to the parking company. If they're entitled to collect money on the supermarket's behalf, that won't be a problem, will it? Alternatively, send the cheque with a covering letter to the Managing Director of the supermarket (copy letter to the parking company). I think there's a lot of scope to play on the contractual relationship between the car park owner and its manager.