Well, of course you can appeal it ! You have the absolute right in law to take the council all the way to the adjudicators, but would, in all likelihood, have to forego the discount option, (see your PCN) if you want to succeed.
You need to be aware that councils ruthlessly game the system to maximise their cash flow, so your initial representations will, in all likelihood, be rejected, thus forcing your to either cough-up, or wait for the next stage, the Notice to Owner. Under the legislation, the vehicle owner has total responsibility for a PCN, so if you are minded to take them all the way, are you the owner, and is your V5C Registration Certificate for the car up-to-date ? The NtO is sent to the name and address on the V5C.