When you get the permit, find a person to give you a lesson in park campfire safety. Although much fire safety is common sense, there are a few specifics. Sand is a safer way to put out a fire quickly than water, for instance (no steam). Water finishes the job and cools it down, so you should have two buckets of sand and two of water right beside the fire, and a shovel to move things around if necessary. In our experience it's best to build the fire on level ground, not dig a pit. That way there's no slope for anyone to stumble down toward the flames. For additional safety you might wish to erect a tripod over the fire. You might also consider having a blanket on hand in case you ever need to smother flames on a person.
In seven years of frequent cooking fires at Dufferin Grove Park, with between 5 and 25 people around each campfire, with school classes and day camps and people who don't speak each other's language, we have never had an accident. That's partly good fortune - unexpected things can happen - but also good attention to safety details.
The main thing is to locate the fire on level ground, with ample room for people to keep a distance on all sides of the fire. There should be no nearby obstruction -- a bush, a wall, a picnic table, a path - that requires people to walk too near the fire to get somewhere else.
Because of its heat, fire carries its own natural incentives for people to stay back and have respect. Once in a long while you encounter a person who seems not to notice their position relative to the fire and gets dangerously close without appearing to be aware of it. Point this out to them if they persist. If they don't respond with greater awareness, or if they actually clown around or enlarge the fire or take out pieces of flaming wood, make sure they leave the fire-site at once.
If they don't listen to you, put out the fire right then. The permit is given on the assumption that the person in charge behaves responsibly. You have to be the boss of your campfire, since the buck stops with you.
Children's curiosity about fire: in our experience, children are very careful around fire, and also very curious. If you feel strongly that children must not be allowed near the fire, it's probably best to have the campfire without children present, or not to have the campfire at all. Otherwise it's too frustrating for the children. We've noticed that when we allowed very curious children to have long sticks which they could poke into the fire, they could experiment safely with us right there watching. They quickly learned what they wanted to know about combustion. Fire safety for children means allowing them to learn under the watchful attention of adults, not barring them from the fire site.
Volunteer campfire safety training: Check your fire safety plans with your recreation supervisor or their designated staff. They may give you some tips from their own experience (campfires or bonfires have been a staff-run activity at park/community festivals for many years). Once you've had your discussion (officially called a "volunteer training session") you should be covered by your park's volunteer insurance clause. Then you're ready to settle the final details with the park staff. Get them to show you a water source and a sand source for your pails. If there's none available for you, you'll have to bring containers of water and sand from home: but every park has a source for both those things that you'll discover eventually.