There are two lots and the one you want is probably not the one you tried.
The beach-front lot on the seaward side of Kamehameha Highway is small — a few dozen spaces — and it fills early, from around nine in season and earlier on a contest day or a big-swell weekend.
Sunset Beach Support Park, across the highway inland, is the larger facility, with restrooms, a field and considerably more parking. This is the practical answer, and it is the one visitors overlook because it does not look like a beach car park.
Roadside parking exists along Kamehameha Highway on this stretch and it is how everyone else does it. Two rules: get fully off the tarmac, because this is a fast two-lane highway, and do not block a driveway or a beach right-of-way between the houses.
Crossing the highway. If you park inland you are walking across a road where vehicles do not always do the speed limit. Do it at a marked crossing where one exists, and hold children's hands. This is a genuinely more dangerous part of the visit than it sounds.
Vehicle break-ins happen along this coast. Leave nothing visible, leave nothing in the boot, and consider leaving the car unlocked with nothing in it, which is what a number of residents do.
Contest days — during the November–December window — make all of the above impossible. If a contest is running, park in Haleʻiwa or Pūpūkea and take the shuttle if one is operating, or come another day.