English/Welsh law and Scots law broadly follow the same principles, but often differ in the fine detail of the wording. Lots of cases are won and lost on that fine wording, however.
Under English/Welsh law if the offence under consideration falls under the Road Traffic act 1988, the plod can require the keeper to divulge the details of the driver. ( A T78 iirc ). Failure to do so is an offence. A reasonable defence would be that there is a pool of authorised drivers and the keeper has no way of knowing. Scots law may be different, and you would have to do some research.
However, the fact that a private parking firm is involved suggests to me that parking has been "decriminalised" ( not strictly accurate, because it was never a crime, just an offence ), and now comes under civil law. If so, the police will not be interested because they have no jurisdiction. You will need to research this also.
If it was me, I would start by making an appeal to the better sides of the possible drivers asking the one to own up. If nobody comes forward I would let it be known that my generosity has run out because somebody has dropped me in the soft and smelly, and henceforth nobody can have use of the car. You never know, the innocent parties might persuade the culprit to come forwards. If/when they do, I would get them to identify themself to the parking company. After that, I would file the matter under S.E.P. ( "someone else's problem").
If the above generates no joy, I would consider carefully whether or not I have a strong enough case to fight them and win. If I fought and lost it could potentially cost thousands in legal fees etc. Failure to pay such a debt could result in bailiffs seizing goods ... the possibilities could be horrendous.
Round here, some lawyers offer a fixed price one-off consultation. You present what you have and they tell you whether you should fight or roll over. You lose the consultation fee of course, but this is nothing to what you would lose if you are flogging a dead horse. This is what I would do if I wanted to fight.