The invoice is as a result of an implied contract. The argument is that you engaged with the contract by parking in their car park being already aware of the restrictions.
The obvious argument sits around how clearly the separation between the two parking areas is and how clearly you understand who you are engaging with. In simple terms, it can be argued that you did all that was necessary in ringing the required number. If the signage isn’t sufficiently clear for a ‘normal person’ to understand that this part of the car park is run by a different company, they’re on thin ice. They’ll say what they can to convince you to pay.
I'm Autistic, if I say something you find offensive, please let me know, I can guarantee it was unintentional.
I'll try to give my honest opinion but am always open to learning.
Mark