When I was there Ketchum and Sun Valley were a complete mod scene. It was hard to find a place to even park to ask questions. I they were parking folks in a big lot (at the ski resort I think) and busing them into town. It was bad enough that I vowed to avoid driving to Ketchum again.
I drove to Featherville and the ranger there said you can park at any of the many pull outs on the USFS road. There were also parking lots at the few businesses in town, but you'd have to have permission to park there long term.
If I was going again, I'd park in Featherville. It is pretty off the beaten path so don't expect to find much more than a diner there. You might check that there isn't anything special going on, they have a music festival near there July 31 through Aug. 2, 2015 and it might be a bad choice for parking at that time.