I see that Martin Lewis advises not paying. The problem with that is the "fine" then doubles. I will just relate my experience for what it's worth.
In a supermarket car park, when I emerged from shopping, my car had a flat tyre. I called on Green Flag who came out quickly and took the wheel off and on examining it found a large hole, too large to fix, and recommended a new tyre. The GF guy then rang round a couple of tyre companies and found one stocking the right size, then took me and the wheel round there (leaving my car on his jack) and got me a new tyre fitted. We then returned to the car park where he insisted on replacing the wheel.
I then got the inevitable "fee demand" as it had been well over the time limit for parking. I paid up immediately with a covering letter stating it was "without prejudice" and demanding they rescind the payment and explaining the circumstances and enclosing a photo of the car in the park with the flat (and hinting that as it had been picked up in their park I might sue them for the replacement cost - only bluffing of course, but "every little helps"). They replied asking for evidence of my payment to the tyre company. Fortunately I had retained the receipt, unfortunately they hadn't dated it. So I took an image and edited a date on to it and sent it off. They then returned my un-cashed cheque (they had obviously put it on hold due to my first letter).
So my advice, if you think it's unfair, would be do pay up on time but mark it as "without prejudice" and stating any circumstances to support your claim.