To have a party there do we have to get permission first? Are they able to reserve a spot for us?

You don't need to ask for permission to have a party :)

We'll is a park is public so no. By the way last week there was a birthday party and I asked the same question and they said that is public. And is first come first serve.

You might want to call the city at one time they were requiring a permit for the covered areas, friend used to work for Socorro he's the one that told me