Hello, I am a solicitor with 20 years experience. I will try to help you with this.
Can you just explain a little more please? What sort of place was the car park (eg supermarket?) which company is running the car park? Can you also juts tell me what parking ticket you paid (how much and for what reason?) and why the company are trying to levy a "fine"?
If you are sure this is a private company and not the local authority or other state agency then you do not need to pay anything at all. You do not need to appeal you can simply write to the company saying that you had paid to park, you did not exceed your time parking, they have suffered no loss and therefore have no right to seek any payment of any sort. If it is a private car park this is matter of contract law and they have no power to levy a fine and as you paid to use the space they haven basis to clam compensation either. You need to be sure about who is running the car park though to take this approach.