That's one of the tightest parts of the highway and full of Do Not Stop avalanche areas. I would plan on parking at Red Mountain Pass and walking down the road. There's a webcam on the pass that shows what conditions are.
Google Maps rendered a location for those coordinates at the top of the Chattanooga loop, which made me assume you planned to start a ski trip up the jeep road to Bullion King Lake, which meets the highway about a mile below Red Mountain Pass. There's no route directly west from Chattanooga up Mill Creek. The road shown on the map goes to a mine and stops halfway up a cliff. That's not practicable even in summer. The standard route to Columbine Lake is the zigzag trail (#509) that starts off a side road that runs north from the Mineral Creek bridge which the jeep road crosses just west of the highway as it starts up toward Ophir Pass. That's pretty steep for winter (but at least no snowmobiles). There might be room for parking in the wide spot at the Ophir Pass turnoff. There's also a large scenic turnoff area on the west of the highway just south of Chattanooga.