Just noticed that Howard beat me to it, but I'll leave my post anyway!
Hi Fii and welcome to the forum. If you have read any of the other threads concerning Lidl and the antics of their agents on these boards, you will see that the parking companies concerned are a law unto themselves.
The good news is that as this is a private car park, even if you had not written to explain the situation, the parking company have no legal entitlement to get you to pay. The
parking ticket relies on a contract being formed between the driver and the landowner (or their agents) that a breach of any agreement formed will result in payment of damages to the landowner. Consequently, having made no such agreement to pay such an extortionate rate (the 'contract' would have to be proven to have been individually explained and understood by you) the parking company cannot force you to pay. The parking notice represents an invoice asking you to pay money, nothing more, and since your parents did not agree to the charge, no action can be taken.
They can, and will write thretening letters saying that they will firstly increase the charge payable, then will tell you that they will 'prosecute' you, and maybe even instruct bailiffs to collect the 'debt'. They may then send the account to a
debt collection agency with further letters to follow.
That is the bit that frightens the average person, and is carefully designed by the parking company to bully you into paying their invoice, but providing that your parents are resilient and not easily scared, this need not be any cause for concern whatsoever. Firstly, they cannot prosecute as your parents have not broken any law (and in threatening prosecution or using the word 'offence' as they often do, they are actually committing a criminal offence under the Administration of Justice Act) - this can only be a breach of contract, for which they would have to take civil action (but they won't as they cannot possibly win). Secondly, if they threaten bailiffs, this is another offence, as they cannot instruct bailiffs until a judgement has been obtained at a
County Court, so refer back to the previous sentence and see this for what it is, a meaningless threat, and harassment in contravention of the law. Finally, if a
debt collection agency was to contact your parents, they only need to state in writing that the matter is in dispute, and the agency cannot lawfully take any further action and would have to refer it back to the parking company.
That explains the process, but now to the matter in hand. Write one FINAL letter to these sharks, along the lines of:-
Reference XXXXXX
Date XXXXXXX
Sir
I write regarding your recent communication in respect of the above reference numbered invoice. Please note that I do not acknowledge any debt to your company. I have entered no contract to pay you the amount shown on your invoice, nor will I do so should you write to me in the future. My vehicle was parked on the landowner's land, with the full knowledge and express permission of said landowner specifically the Store Manager acting with the full authority of Lidl Stores Ltd, and in doing so any action taken by yourselves on behalf of the landowner is null and void. Confirmation of my arrangement with the landowner has already been communicated in writing to yourselves and it is in that direction that any further correspondence on your part should be sent.
In the light of this information, I demand that you cease and desist any further harrassment or attempt to demand money in respect of your invoice, with the exception of a written confirmation that you have ceased all action in respect of this matter.
Yours,
XXXXXXXX
Get that off recorded delivery, and keep a copy of that and any previous correspondence. It may well be ignored, but then so should any future correspondence from them.
Brocken down vehicle parked in a disabled bay with Mgr of Lidls permission 
I would not pay a penny;
