When I contacted the National Park Service, they had this to say about their "commercial truck restrictions":
Is this vehicle registered Commercial? And YES we do have height restrictions. We have bridges under passes, which would not allow anything over 13 ft.
I asked where the 13' overpasses were but didn't get any response.
IMO, stick to I-95 instead of 295 (Baltimore/Wasington Parkway).
There are no height restrictions with the Francis Scott Key Bridge. You have to take it or the topside of I-695 around Baltimore since you can't go through the Baltimore tunnels with propane. I prefer it over the top side.