You certainly can! There are two (2) rounds to fighting a parking ticket. The first round is the Dismissal Round. If you lose on that round you may or may not have to pay the ticket. For example, in New York City you DO NOT have to pay a ticket while it is being Appealed (the second round). Either way, you can mail in a second request for dismissal called an Appeal. In actuality, you are appealing the first Judge's Determination.
The Appeal does not go to the first Judge - it goes to a different Judge. If the second Judge agrees with the first Judge, that's it. You don't owe any more money because you have already paid for the ticket. But if the second Judge disagrees with the first Judge, then the first Judge's Determination is reversed. That means you get a refund from the municipality.