There's a spot about half a block away across the street called Benny's liquor. It's a nice size parking lot and it shouldn't be a problem early in the morning.
Azusa Off Road Park - SCCX Day - 8/25/2012
There's also a Target near the corner of Azusa and Foothill.
The parking lot at the top of Azusa usually gets full with road bikers and people out for walks.