Sounds like you really need to explore a Plan B for days when you are sick or the roads are too icy for your road bike (although there are ways around the latter... as I'm sure you are aware).
Right now it's the car, but in future it could be transit.
Before you sell your 2nd car, take some time to explore just how bad transit it on the weekend. I frequently hear people -- in many cities -- tell me how bad transit is. But many of them have never really checked out prices, routes and schedules. Also check out cabs and compare the cost of an occasional cab to the cost of insuring, parking and maintaining the 2nd vehicle.
Many folks who bike commute and use their cars as backup aren't always aware of options. And there may be others in your case.