We try to site see by leaving our rig at the somewhere and taking public transportation.
For example Nashville has 2-3 Walmart's with good public transport to get you downtown.
Natchez has the convention bureau where you can leave you rig and take a cab to the historic section.
Western Chicago area has a couple of parking lots at the METRA station that are big rig friendly.
Rockville MD has a place to park your rig and take their public transport to Washington, DC. You have to be gone by midnight
I have left my rig at a Walmart near SEcaucus NJ and taken the Bus to New York Port Authority bus station for 2-3 dollars each way.
Calling the local public transportation authority will usually get you some ideas