I have asked several times about parking along the road -- during weekends, since construction is only Monday-Thursday, and so on. But I keep getting the same answer: NO parking along the road period. The no-parking rule will extend for several miles, so walking part of the road effectively wipes out your chances of a successful day hike.
Consequences of parking along without a pass can be that your car will be towed, along with an expensive fine.
Another option would be to get a spot in the Whitney Portal campground the night before and night after, just to use the parking spot.