If you enjoy hikes / scenery, a quick hike at Runyon Canyon Park in Hollywood is highly recommended. This is along the same range as the Hollywood sign, but a couple mountains over. Apparently a buzz place for celebrities as well. Making it to the top will reward you with sweeping views of Downtown LA, the Hollywood sign, and general prettiness. It was starting to drizzle when I went up, and you can distantly see a rainbow in the middle of the above picture.
You can try finding street parking in the residential neighborhoods nearby, which I lucked out on. There’s everything from a steep ascent to a gradual incline that twists up, so you can pick what you’re in the mood for.