You have been served a Penalty Charge Notice by the council who are enforcing parking contraventions using the civil law, under an Act of Parliament called the Traffic Management Act 2004. The process of enforcement is defined in that law and its associated regulations.
The key feature of the process is that the owner of the vehicle has full responsibility in law for appealing or paying PCNs, the person driving the car is irrelevant. Of course most times the owner is the driver.
It would be best if you submitted an informal representation to the council explaining the circumstances and ask them to cancel the PCN as they have lost no revenue at all. You can prove payment. Do not be confrontational as the reason for the PCN is your error.
Post up a draft of your reps here for review before submitting, then after submission, post up their response when you get it. Don't be shocked if they refuse your reps. Most councils reject all informal reps, because they know that if they do, the appellants just cough-up the money, to get the discount and not being "au fait" with the process an the legislation.