Usually the host pays. If it's at a sit-down restaurant with a menu (where different food have different prices and people can order sides ans dessertand drinks and stuff) everyone pays for themselves... but surely you're not having a kids birthday party at a sit-down restaurant? It's a place that specialises in servicing children and hosting child parties, where you can order a buffet for everyone/the kids, right?

But as long as you communicate expectations clearly on the invite you can do as you like. For example: "There will be buffet style food served for the kids to eat. Accompanying adults can buy food at a café next to the playground." Or "Entry to the jump yard is X:- per child. Since each family pays for themselves we don't expect birthday gifts" (make sure your child knows this).

Parties can be made pretty cheap, so I would prefer a type of party that fits your budget rather than create an expensive party where some kids might not affort to join the celebrations. It's not the location or fancy foods or special activities/entertainment, it's the people and (for many kids) the theme, that makes it special.