I see that this park closes at about 7:00 p.m. or dusk. Is there a place nearby we can park for walking on the beach at night?

Right next to you... go left, & loo left...parking is free until 10pm

I think best to park in the beach parking spots on the many street that end by the beach. We parked at the end of Madison last time parking at park was full.

Facing the beach, on the right hand side, there is parking outside the park on the side street and another street over. I can't say for certain that those aren't also timed but they seem more residential.