I think 8 year olds can be supervised by the party hosts in a pizza place with a small arcade. It's contained, and they are used to being in school in large groups and without their parents.
I would just put a "drop off" and "Pick up" time on the invitation rather than an open window of "12-2 PM" or whatever the hours are. Greet the parents, and say welcome to the kid, then say to the parents "Thanks for bringing him, we'll see you at 2 PM. Kids will be in the arcade for part of the time, supervised by my husband and me, but the pick-up will be in the party room."
If you want to put on the invitation that the kids will be given $3 in quarters (or whatever the amount is) for the arcade and state you'll be serving cupcakes and to let you know if there are any dietary issues/allergies, it's clear what the party activity is.
Beyond that, any parent who stays will clearly pay for food for themselves if they choose to eat. I don't think you have to explain yourself beyond that.