I have not seen parking discussed since the Whitney Portal road construction project was completed. The overflow parking lot has always been available to long-term parking, except in 2016 when the road construction made parking difficult. Since the completion of the road construction, I am pretty sure you can park along the road, if the overflow lot is full.
But I am not sure. Please call the Wilderness Permit office number, 760-873-2483, and ask about the roadside parking. And please report back what you find.
There is also the option of parking along Minaret Road (in Mammoth) just below the Mammoth Mountain Inn. It puts your car in the middle of the JMT -- a little easier transportation-wise on each end, since you only need to use one bus shuttle at each end.
