I was checking the 'Protection of Freedoms Act' part to see if there was a technicality you could exploit as a get out of jail free card in case Aldi were of no use - alas there isn't. As such, there's no need to hide the identity of the driver any more, as they can hold the keeper liable anyway.
In any case, from what you've described then, it certainly sounds like there'd be a decent argument for 'frustration of contract'. If you're not familiar with the concept it may be worth doing some research into it, including on the PePiPoo and MSE parking forums. The concept is essentially that an unforeseen event, out of your control (i.e. breaking down), which happened after the contract was formed*, makes it impossible for you to comply with the contract (i.e. to not remain in the car park for more than 90 minutes).
Depending on how stubborn ParkingEye are, this may need arguing in court, but on the face of it you'd seem to have a good case. Definitely get on to Aldi and ask them to intervene, but keen an eye on the deadlines - if Aldi haven't resolved it, be sure to submit an appeal before ParkingEye's appeal deadline. You can share a draft here for comment before doing so.
*This is why I was asking about the circumstances - we've seen cases where people have noticed a fault in their car, then pulled off the public road into a private car park and waited there for recovery. In those cases, it's harder to argue the event was outside of their control, and that it happened after the contract was formed, as they were aware of the issue when they parked. In this case, it seems the driver was not.