If you can evidence that you didn't receive the original letter, you should be able to appeal the fine outside of the usual window. If it's been a year since the incident, you may even be able to argue that it's unreasonable for them to ask you to pay it at all. How you'd prove this though, I'm not sure. Were your contact details correct on the copy of the original letter they gave you?
Submit an appeal in writing anyway (you can find templates online). They should reply to this within 56 days. Usually if they don't, you automatically win, however I'm not sure if this will be the case for you outside of the appeal window.
If they reject your appeal, you can make a formal appeal to either the Independent Appeals Service or Parking on Private Land Appeals. You'll need to find out which one the company is a member of.
You say that you've received court paperwork...is this an actual summons to attend court? It would be very unusual for this to be the case if they've only contacted you twice about the fine.
Ultimately if your appeals are rejected and they do decide to take you to court, you'll have the opportunity to present your case. It's worth considering though that if you lose, you'll be liable for court fees, legal costs etc on top of any fine. You'll end up with a CCJ which will affect your credit score for years. With a CCJ, they can also instruct bailiffs to attend your property and remove items of value if you can't/won't pay.
You should obviously take legal advice if it gets that far, but given your lack of evidence, paying may be your best option. They will almost certainly be willing to set up a payment plan.