How about sending...
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I was surprised to receive a Parking Charge Notice from you stating that I owe you money for "Whole Period Of Perking Not Paid for", as the whole of this parking period was paid for and I have retained the ticket issued at the time which proves this.
I attach a scan of the ticket your machine issued which shows that £1.50 was paid at 09:33 on the 14th June. As the vehicle left the car park at 10:56 (as your letter states) but the ticket covered parking until 11:33, no further payment is due.
I note that despite me entering my full registration twice into your machine, the ticket I was issued only shows the first part of my numberplate, which may be why you have not correctly allocated the payment you accepted.
I look forward to receiving confirmation that you have cancelled this Parking Charge Notice.
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...and then come back here if that doesn't get the job done?
I'm familiar with the Council PCN process from Pepipoo and here, but I don't really know my way around the private parking side - am I right in thinking the 'only one bite of the cherry' rule that prevents councils changing their mind about the reason for a PCN doesn't apply here, and that this might have to go to POPLA instead of the adjudicators, who might not be as persuadable in their interpretation of the rules?