Thought I'd give a final update...
Talked to a manager at the restaurant where I was ticketed. He said to bring in the ticket and he'd reimburse me on the spot.
About a week later, I was finally in the neighborhood again and ran in to take care of this matter. There was a different manager on duty than the one I'd talked to.
Initially, he said he'd have the valet company cut me a check. When I mentioned that the other manager said he'd pay me cash on the spot and had suggested I pay the ticket first so they didn't increase the fee, he agreed to give me cash. I also asked about getting my money back for the valet fee, too... I wasn't going to fight too hard, but figured it was worth at least asking. Got that, too. So they stepped up and dealt with the situation pretty well... thus I will withhold the identity.