You say in the thread title that the car park is not owned by Lidl. Is the car park shared with other stores, like a retail park? The fact that Lidl may not own the car park, is not the issue. It is who contracted ParkingEye to operate the parking there.
Do you have a receipt or evidence of any Lidl purchases on the day? Plan A is to always try and get Lidl to get the PCN cancelled. If you tried and were robed off by some lowly customer service bod, then aim as high up the management food chain as you can go, starting with the CEO, Ryan McDonnell.
Email him at ryan.mcdonnell@lidl.co.uk and explain the situation and tell him how you are unhappy with being charged £100 by a third party simply for being a valued customer and that they should get their contracted parking operator to cancel the PCN.
Include evidence of being a customer that day.
Don't appeal to ParkingEye until you've heard back from your complaint to Ryan McDonnell. If you've not heard anything back by Tuesday 8th July, then just follow this advice to send an appeal to ParkingEye:
There is no legal obligation on the
known keeper (the recipient of the Notice to Keeper (NtK)) to reveal the identity of the
unknown driver and no inference or assumptions can be made.
The NtK is not compliant with all the requirements of PoFA which means that if the
unknown driver is not identified, they cannot transfer liability for the charge from the
unknown driver to the
known keeper.
Use the following as your appeal. No need to embellish or remove anything from it:
I am the keeper of the vehicle and I dispute your 'parking charge'. I deny any liability or contractual agreement and I will be making a complaint about your predatory conduct to your client landowner.
As your Notice to Keeper (NtK) does not fully comply with ALL the requirements of PoFA 2012, you are unable to hold the keeper of the vehicle liable for the charge. Partial or even substantial compliance is not sufficient. There will be no admission as to who was driving and no inference or assumptions can be drawn. ParkingEye has relied on contract law allegations of breach against the driver only.
The registered keeper cannot be presumed or inferred to have been the driver, nor pursued under some twisted interpretation of the law of agency. Your NtK can only hold the driver liable. ParkingEye have no hope at POPLA, so you are urged to save us both a complete waste of time and cancel the PCN.
This will give more time for Lidl to respond. Any initial appeal will be rejected, no matron what grounds. In this case, ParkingEye will write back as king for the drivers details, which you can ignore and eventually they will reject the appeal and give you a POPLA code where you can appeal with all the information you provided and the Equality Act 2010.