The usual argument with breakdowns is "frustration of contract", that is, you were unable to comply with the terms on offer due to matters outside of either party's control.
One consideration here however, is that the notice from ParkingEye suggests that if you're going to stay for longer than 2 hours, there is the possibility to pay for parking, and ParkingEye will seek to argue that once the driver knew they were going to be there longer than 2 hours, they should have paid.
As a starting point, the keeper should speak to the service station owners/management (not telling them who was driving) and see if they'll intervene. If they won't we can pursue appeals.