For clarity, as the notice is addressed to your partner, the appeal must be done in their name. They should appeal as the keeper only. H C Andersen's appeal above covers the main points, and I agree with brevity. I personally take a slightly different approach in that I think it's worthwhile mentioning the relevant legislation, as it helps demonstrate to the operator that you know what you are talking about. Here's my usual template appeal for these cases:
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 note from your correspondence that you are not seeking to hold me liable as the registered keeper, under Schedule 4 of the Protection of Freedoms Act 2012 ("The Act"). You have chosen not to issue a Notice to Keeper in accordance with The Act, and it is now too late for you to do so.
There is no obligation for me to name the driver and I will not be doing so. 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 28 days, chase them.