That NtK is not PoFA compliant. Whatever you do, do not identify who was driving, inadvertently or otherwise. They have no idea who the driver is and you, the keeper are under no legal obligation to identify the driver. Because they are not relying on PoFA, they cannot transfer liability from the unknown driver to you, the keeper. They cannot assume or infer that you must be the driver because you are the keeper. So, do not reveal the drivers identity.
What was the reason for you visit to that car park? Were you a patron of a business/shops etc.? If so, have you tried Plan A, which is to ask the business/shop/landowner to get the PCN cancelled?
You should also prepare an appeal for Plan B, to Gemini. Exhaust Plan A first but don't miss the appeal deadline. On or before Tuesday 11th June, send the following (verbatim) if you have not managed to get it cancelled through Plan A:
I appeal as the registered keeper. I am not obliged to identify the driver and decline to do so. You cannot transfer the driver’s liability (if any) to me as you have not served me with a Notice to Keeper that complies with Schedule 4 to the Protection of Freedoms Act 2012 (PoFA) and it is now too late to do so. In particular (without limitation) your notice does not contain the warning required by PoFA paragraph 9(2)(f).
As there is no legal presumption that the keeper of a vehicle was its driver (as opposed, for example, to being a passenger) on any particular occasion, Gemini Parking is unable to pursue me as driver. I now require you to cancel the parking charge and remove my personal information from your database
You need to understand though, you are not dealing with an organisation that has any sort of customer service ethos. This is a small firm of ex-clamper thugs out to scam you for as much as possible. Not only that, but their Accredited Trade Association, the IPC, are equally noxious and their secondary appeal service, the IAS, is a sham. So, if your appeal above is rejected, you can try a secondary appeal to the IAS but with less than 4% of all IAS appeal being accepted, you are likely wasting your time.
Lets' see what the response is to your Plan A and Plan B first.