You might want to suggest using one of the public parking lots near Sunset and Vermont, or anywhere along Hillhurst and taking the DASH (Observatory) bus into the park. Super convenient! Also there is usually ample parking up Fern Dell Road leading to Western Canyon Trail. Also for "Batman" fans, the original Bat Cave is right on the Bronson trail hike. :-D Love this video!
There is also a great hiking trail that goes from near the merry-go-round up to the top of Mount Hollywood, not to mention the great trail through Fern Dell.