Mid week it should not be a major issue. Its timing issue, 8 AM is bit early to get spots from folks staying at the hut heading down after breakfast. Dayhikers tend to fill in the slots from about 7 AM to 9:30 AM, the hut guests tend to head down Valley Way and get to the parking lot around 10 AM and then slots open up as the day progresses. That is the reason why in the evening after day hikers have headed home that there will be cars parked along RT2 well away from the parking lot while there are empty spaces in the actual lot.
If the lot is full the key is to park completely off RT2 preferably completely off the pavement. The gravel shoulder is quite wide. A short distance to the west, about 500 feet across from the tennis courts is a rough spot that will hold ten cars completely off of RT2.
Or you can do what the locals do and park over on Durand Road at the original trailhead for the trails up the Northern Presidentials at the Ravine House site. Its a short walk via the Beeline around the west end of Durand lake to Appalachia or across the dam by the pool and up by the Tennis Courts to Appalachia. There is short dirt road on the south side of Durand that is shaded by trees, makes a big difference in how hot your car gets while parked.