I got there around 7:30 AM yesterday (a Saturday for future reference) and there was just one other car in the massive lot. I passed the lone female hiker about 1/3 of the way up and had the trail all to myself the rest of the way. While on the tower platform, a group of three men (whose conversation led me to believe they were also keeping their fire tower completer credentials up-to-date) arrived and waited for me to descend the tower before going up themselves. During the hike out, I believe I passed two other small groups. So not bad. Early is clearly key.
I was also struck by the size of the parking lot. The article says 40 cars, I think, but it can probably hold 100, IMO. But then again, the Fire Tower Challenge is quite popular so the size is probably justified.