"Parent and invited child only, please." Or as others put it, "Due to limited space at Chuck's, we don't have room for siblings -- one parent and the invited child only, please."
If someone ignores this and brings extra kids: You or your designated adult (don't do it yourself, have a close and tough adult friend do this job) says, "Hi, you're here for Little Man's party? Your name? Great. I see you've got a couple of kids with you -- who's Sam's invited friend? Sally here? Nice. Sally and you as the parent are part of the party, and that includes pizza and drinks at the party table, but other kids will need to have entry tickets and their own table if you want, and you can get that right over there at the counter." Or whatever's appropriate for the setup.
In other words, yes, a kiddie bouncer, who with a big smile lets the parent know that he or she and one invited child are paid for but the other kids are not and are not welcome to sit down and chow on pizza at the birthday table. The job should go to someone who isn't afraid to say to a parent, "We're glad all the kids are here to have fun but the party itself is limited to the invited child and one parent as the invitations noted."
I hope you don't have to go this route but if you're really concerned about this happening despite however you word the invitations -- I'd truly have someone prepared to sort the groups as they come in. You might not have to do it at all but be prepared.
It think the families who brought multiple kids to the earlier party were way beyond rude and thoughtless. I doubt you'll have the same issue at a place like Chuck's where most parents generally know that it costs the hosts per person.