Your next parking opportunity is after you cross the RR tracks, taking a left onto Sutton (not very popular with traffic behind you however, as there is no left turn light.
Let's say you continue up Broadway after crossing the tracks. And if you so choose, there are a couple of lots off of Broadway, which would have been to your left after crossing those RR tracks. But some of them are specific for the restaurant or business, as designated by signs on each parking spot.
You have a choice at the next light (State St.) to go left, and look for one of the few on-street parking places (on the right side of the street only).
Or maybe you''ll chose to go right, at that same light, here...
Now you will have the great opportunity to drop by the visitors center of course.
Take a left at the first signal light.
Which is right next to a brand spanking new parking lot as part of the Town Square.
Since this was a rainy day, not as many folks were walking around downtown Black Mountain as holidays often have. But I think you've got lots of opportunities, and it's not too bad just driving slowly around until you see where you want to visit. Be warned, you will drive slowly!
Did I mention that all this parking is free?