I love doing that route, but I start off from Arastradero Road. Arastradero Road connects Alpine road and Page Mill Road, there is a parking lot in the middle and also a parking lot at the very end where it meets Alpine road. The parking lot in the middle usually gets full on the weekends if you get there too late, but there is usually plenty of parking at the second one I mentioned.
Check googlemaps streetview to see what I'm talking about more clearly.