Personally, if I get a parking notice from a private parking firm I ignore it. I've probably had a dozen from Excel and Britannia Parking. At the most it led to a few phone calls from some pseudo-solicitor asking for money, but more usually just a few letters increasing in threat. I blanked them at every turn and they eventually went away. The people I work with have had hundreds between them over the years, and do exactly the same as me.
Obviously it's up to you what you do, but on a sheer point of principle I wouldn't pay a fine* when I've just driven through the place - it's utter nonsense. Nor would I trust them to be reasonable when you explain your situation.
However, it's not pleasant having them chase you, and I wouldn't be so presumptuous to advise you to do what I did.
Good luck.
*they're not fines, they're invoices. Only the plod and local authorities can fine you.