Please tell us that you didn't identify as the driver when you communicated with LNER. Any communication from you must only be as the Keeper of the vehicle. If you have identified yourself as the driver, you have blown away any defence should they be so stupid as to try and prosecute this obvious error with there ANPR system.
Looking on GSV, LNER appears to manage the car park themselves but I cannot read the detail on their signs, or whether the logos on the bottom reflect any affiliation with an ATA.

Depending on the wording on their signs, they will either issue a Parking Charge Notice (PCN) under civil contract law or a Penalty Notice (PN)under railway byelaws as a criminal offence, if they decide to try and do anything about it at all.
You need to demand clarification from LNER about their intentions of this obvious failure of their ANPR system and a "double dip" scenario.