Looking at the notice - they have not complied with the requirements of Schedule 4 of the Protection of Freedoms Act (there is a link to this legislation in my signature at the bottom of this post) - even if they otherwise had, as this is railway land it is likely not to be 'relevant land' as defined by the legislation. On this basis, they are unable to recover the charge from your wife as the registered keeper of the vehicle. At this stage your wife could therefore appeal with the below:
Dear Sirs,
I have received your Parking Charge Notice (Ref: ________) for vehicle registration mark ____ ___, in which you allege that the driver has incurred a parking charge. I am the registered keeper but I was not the driver - there is no obligation for me to name them and I will not be doing so.
As the car park in question is railway land that is subject to byelaws, it is not 'Relevant land' as defined by paragraph 3 of Schedule 4 of the Protection of Freedoms Act 2012 ("The Act"). As such, you are unable to recover the charge from me, the registered keeper.
Even if the car park is relevant land, which is denied, your notice does not comply with the requirements of Schedule 4 of The Act, and you are therefore unable to recover the charge from me, the keeper.
I am therefore unable to help you further with this matter, and look forward to your confirmation that the charge has been cancelled. If you choose to decline this appeal, you must issue a POPLA code.
Yours,
If appealing online, be careful there are no drop down/tick boxes that cause you to identify who was driving, and keep a close eye on your spam folder for their response. If they do not respond within 35 days, chase them. Your wife should be the one whose details are in the appeal, and she should be appealing as the registered keeper only.
