If Premier Park want to be able to recover the charge from you as the hirer, they need to comply with the requirements of Schedule 4 of the Protection of Freedoms Act (there's a link to it at the bottom of my post in the signature) - in particular, paragraphs 13 and 14 are relevant in the case of hire vehicles. If you give that a read, you'll see that 13(2) requires them to provide certain documents alongside the notice to hirer, which they pretty much never do.
For the initial appeal, you could include something along the lines of 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 appealing as the hirer of the vehicle, on the grounds outlined below:
1. The alleged contravention did not occur:
The notice alleges that the charge was issued for "Parking in disabled bay without clearly displaying a valid disabled badge". This is not the case. The driver at the time was not parked in a disabled bay, but was instead parked in a 'Parent and Child' bay, which they were entitled to do as they were travelling with a small child at the time.
2. No hirer liability:
I note from your correspondence that you are not seeking to hold me liable as the hirer of the vehicle, under Schedule 4 of the Protection of Freedoms Act 2012 ("The Act"), as you have chosen not to issue a Notice to Hirer in accordance with the Act. There is no obligation for me to name the driver and I will not be doing so.
For the reasons outlined, I am not liable for this charge. 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 very 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. Only choose 'Hirer' if it asks for your relationship to the vehicle, no blabbing about who was driving. If there's a drop down asking your reason for appealing, choose 'Other'.
If you're not likely to be going to the site to get pictures any time soon, you could just submit with point 1 removed, and save it for your POPLA appeal if they reject your appeal.