Sorry to hear of the issue. You clearly have evidence to support your defence so my view is you continue to refuse payment - if they sued you, I believe your defence would succeed.
A parking fine results from breach of contract - either you didn’t pay when you used a car park, or you did but you overstayed. The fine is fairly low initially but will increase for as long as it remains unpaid - with debt collection charges so it can easily double. Whether you pay or dispute the fine depends on the facts.
You should reply to them to set out why you dispute the fine - the pre action protocol does state that parties in a dispute such as this one should communicate and try to avoid litigation if possible. You could make them an offer "without prejudice to liability" which means you are offering a sum whilst making no admission of blame - you do not have to this though. They may or may not accept an offer.
Debt collectors are not bailiffs - they have no powers as such. They will simply try to collect the money their client thinks you owe. You can turn them away if you dispute the debt and they visit you - they can’t enter your home and take goods. They would report back to their client they were unsuccessful and it’s then up to their client whether to take matters further.
They should send a letter before action to you if their debt collection attempts fail - which is required under the pre action protocol to give you the chance to avoid court action. They usually give you 14 days to pay before they will take civil court action.
If they do sue, you will receive court documents (a response pack) which you must complete and return to the court. It is a tick box exercise for the most part and there is a short section to write a defence which is easy enough though please feel free to come back to this site if you need any more help.
You should not ignore the court papers as if you do, the claimant will apply for judgment in default meaning they win and you would not be allowed to defend the claim.
A claim will also take 9-12 months to be decided at court. If you lost then you would get 14 days to pay the judgment before the claimant can enforce the order, and 30 days to pay in full before it is registered with credit agencies. The claimant cannot recover legal costs if they win, in a claim under £10K (a small claim), all they can claim are the court fees and interest.
If the claim has no merit (and it sounds like it would have no merit given your evidence) then you have an option (after your defence is filed) to apply to strike the claim out. The court will consider an application if the claim has no merit, or is misconceived.
You should first invite the claimant to voluntarily discontinue their claim within 7 days if it gets to this point - tell them if they don't then you will apply to strike out and seek your costs if you are forced to do so. If you have no response or they reply and refuse then you could apply to strike out.
The application costs £275 but this is recoverable if your application succeeds. If you are on a low income, have low savings or in receipt of benefits then you can ask the court to waive the court fee. If you won the application, the claim is struck out.
The hearing (if the case gets that far) is likely to be held remotely, it's you, a district judge (who is a practising solicitor or barrister) and someone from the claimant company. The Judge decides and if you lost, you get 14 days to pay the sum. If paid in full within 30 days then nothing goes on your credit record. But certainly I think you would succeed in a defence, just to be clear.
I can assist you going forwards if it gets to the point they issue a claim. Note that nothing will go on your credit record - if you lose, you have a full month to pay before anything would be registered on your credit record.