Wow, get ready for just the first of many birthdays to have to navigate how to avoid spending a FORTUNE. It can easily happen but don't let it.
Almost any kids' bday party other than your home involves some kind of activity for the kids that you have to pay per child. You are NOT obligated to pay for the parents, and it is NOT expected unless you run with quite wealthy people. Middle class people can't afford that and if you are friends w/ other similarly middle class people, they won't expect you to. The parents can always participate (and pay) for themselves if they want, but myself, I have taken my kids to bowling parties and everyone knew they were for the CHILDREN not the adults to partake in free bowling. I would never even think of such a thing unless it was a close family member and we as a group wanted to play each other due to our family relationship. Many hosts WILL pay extra so the parents can have pizza too, and this can get expensive as well, so I usually would pay for a certain number of extra pizzas to allow each parent a slice or two, and also offer them cake, and that is as far as you need to go.
Don't stress over it. Just explain when they arrive that the kids will be on a whatever lanes and whatever else is included. They'll get the point, and if they are confused and new to parties, they too will soon learn the script as they attend more as their kids get older. This is one rare example of when you are grateful when your kids get older and the parents just drop their kids off and you can avoid parents costing you more when they eat.
And for those who can afford $500 to pay for "everyone and their momma too" don't lecture those of us who can't. Kids parties are for kids!!
OK, I am editing to add this - in our family what we do is have a "kid party" often on the weekend in which it is for the kids (as described above) and on the actual birthday evening, the family goes out to eat wherever my child wants to go, and everyone pays for themselves but brings gifts. After I serve everyone cake. Our family members all do it this way. If you are mixing family and friends at the bowling party, I understand it could be more confusing if you and/or your adult family members WANT to bowl too, but yet not pay for the other friends' parents. So I decided to add what we do in our family. As your son gets older and makes more friends, this "method" can save you a lot of money. The scenario I first described above is really for classmates and kid friends, and is a little harder to "enforce" when many attending are family w/ their own kids of various ages (like if you pay for the little ones do you have to pay for the almost grown teens, etc.). That's why we keep the kid party separate (but still invite similar age cousins, etc.) and the family dinner at another time.
Hope this helps!