How can I take a permit for barbecue if I just went with my family of 3 ?

I don’t think they require a permit for the picnic tables scattered around the park, unless you want to use the gazebo. They do have regulations you have to follow though posted at the park.

You shouldn't need a permit for just a small group of 3

You don't need a permit for a small bbq. Permits are needed when you want to reserve tables for something like a party