I lived in Santa Barbara for 4 years and you definitely don't need an Adventure Pass to park along the paved portions of Camino Cielo, including the Cold Spring trailhead at Cold Spring saddle. (short version hike)
I recently went back to the SB area for a backpacking trip and we left the car at the saddle for several days. I checked at this time with the Los Padres rangers and they confirmed an adventure pass is not needed here!
So save yourself $5 and enjoy!!