Do they need to prove it was parked?
Their more fundamental problem would be proving who was driving the car, in the absence of legislation allowing them to hold the keeper liable.
ANPR-based systems tend to take a picture when the car enters the car park and when it leaves. Whilst that doesn't show that the vehicle was parked, and the burden of proof rests with the claimant, unless a defendant could offer some evidence to show that it
wasn't parked, evidence of a car entering a car park, spending a sufficient amount of time there, then leaving, would generally be enough to show that on the balance of probabilities the vehicle was parked during the intervening time. There are exeptions of course, but they're of little relevance here as this is a Scottish case and therefore will not be going to court.
