They don't need particularly strong grounds to offer an out of court disposal - no matter how many times you ask the question.
We mostly apply the law to the facts. The facts in this case being that your Mum may or may not have hit a parked car (and if she did, she wouldn't have noticed because of the dogs in the car which were distracting her.
If she hit the other car, then on the face of it her driving fell below the standard of a competent and careful driver - so the offence is made out.
If she didn't hit the other car, then despite what you have told us, there is no objective evidence that it did.
This leaves us with strategy as regards likely evidence and the strength of such evidence.
If they filmed your Mum hitting their son's car, and give evidence to that effect, and adduce the recording as physical evidence, it is difficult to see how your Mum would be able to defend the charge.
If they merely claim to have seen your Mum hit their son's car, and your Mum's evidence is that she's not sure because she was being distracted by her dogs, unless you can cast doubt over the veracity of their evidence.
If you can prove that they made a fraudulent insurance claim to include pre-existing damage, you might be able to cast doubt, or they might not want to give evidence.