You can park free of charge in Clovelly Road directly at Clovelly Beach. The beach is not so famous and therefore not as crowded as other beaches – especially during the week days. You have showers there, but only with cold water, but they are still ok. There are of course toilets and a beach bar is not missing either.
It’s a nice place to ride big waves due to the special bay this beach forms, have a look at the pictures! And you can find places in the shade until around 2pm.
The beach is right by a hidden and off-the-path residential area and you should be fine to spend the night there. Just don’t bother the residents there (as usual).