No guarantees, but if you are persistent, and stalk the recreation.com web site you may very well get the needed permits. Also, if that fails you can likely get "walk-up" permits. Aug. 1st is a Monday, and normally there are walk in permits available as cancelations come in, and as group sizes shrink. Parking may be an issue, but it could be that by Aug. 1st the top portion of the road will be completed, and it is possible that parking will then be available. Who knows what the county will do (I certainly do not).
For instance, I just checked, and there are 4 available for June 28th.
