If you're staying at the Claremont Hotel, you should have no problem with parking (or even if you're not, you could inquire about paying to park in their lot).
The best driving might be along the top of the hills behind the Claremont Hotel -- the road is called Grizzly Peak. Take Claremont Avenue up into the hills. At the top, turn left onto Grizzly Peak Boulevard and drive as far as you want. Go back down the way you came up.
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