Yes, it does fill up quickly on Saturday afternoon. Remote parking is about 20 minutes for healthy adults & 40 minutes with small children.
Yes they do and yes there is parking down from the park that you can walk from. They fill up fast as well.
It is a beautiful and busy beach but there is always a place to park. There is plenty of spots and you are never left behind. I will advice you to buy the pass if you are stying few days or planning to revisit de Beach, because the pass let you go first and you don't need to stay in the line for hours.
We went on a Saturday and yes, the lots filled up fast. It was our first visit to the beach there, so I don't know about the other locations for parking.
Thanks! Your answer is awaiting moderation.