As a general rule, where there is a dispute over money, corresponding in writing is generally preferable to by phone, as you have a record of exactly what has been discussed.
I would ordinarily recommend pointing the misread out but as this is Smart Parking, the quickest way is probably a template appeal. Can you show us the notice you have received?
It will almost certainly not meet the requirements to hold the registered keeper liable, and if you appeal as the registered keeper with the below, they will almost certainly cancel the charge.
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,
Do please show us the notice before you appeal, I don't like recommending an appeal without double checking it is appropriate. You'll need to use a third party site like Imgur to upload the notice.
You could if you wish add a line into that appeal before the main point, pointing out that it is not your vehicle and their faulty ANPR has led to them accessing your details without reasonable cause.