Hi, thanks for your question. My name's XXXXX XXXXX I'm going to assist you with it.
Private companies can't punish individuals with parking penalties. The original notice would have been a private land fine based on an allegation of breach of the terms of a contract formed when the vehicle was parked, along with an offer to settle for a specific sum in order to avoid a civil claim for damages.
The registered keeper of a vehicle is now liable for such a breach if they don't disclose the details of the driver. This simply means that the claim can be brought against the registered keeper of the vehicle, making it easier to identify a defendant.
If correspondence relating to a private land fine is ignored, the chances that they will actually bring a claim are fairly low. There are (of course) no guarantees but it would not make financial sense for them to bring a claim in every case. If they did, the amount of the claim can only be for the sum of the fine and a small fee for issuing the claim. They have no claim for legal or debt collection costs.
To be successful in a claim they would need to show that their signage was adequate to create a contract and to incorporate the relevant terms that they allege were breached (e.g. rules relating to markings etc.). Crucially, they can also only claim for the loss they sustained as a result of the breach and they must prove this loss. This amount will not necessarily be the same as the figure they have determined as a penalty and is probably much smaller.
In additional to any internal appeal, if the company is an Approved Operator of the British Parking Association (which can be checked here: http://www.britishparking.co.uk/Approved-Operators), a free appeals process is provided by the Parking On Private Land Appeals service (POPLA).
However, as I say, the charge is not an enforceable penalty that has to be appealed or otherwise challenged by you. It is perfectly permissible to put the parking company to proof on liability and damages for breach of contract by waiting to see if they issue a claim. If a claim is brought it can either be defended or liability could be admitted in part by offering of a small amount to compensate their loss (on the basis that the full fine itself is not representative of such loss and is punitive, unfair and unreasonable).
I hope that's helpful. Please let me know if I can clarify anything.