I vaguely recall that in one of the first threads I posted to here, parents were accused of being 'rude' for _staying_ at a party at a 'paint your own trinket' place, since the location charged per attendee. Evidently there is no sure-fire way to please everyone on this subject!
If that happens, adult pays for themselves.
So now, as a parent, you'd be buying the birthday child a gift and
paying for your own food at Chuck E. Cheese's or painting pottery? So instead of just thinking "Hey, my kid will want to go to their friend's birthday party, how fun" you would also have to consider your budget. "Do I have money to purchase pottery that I don't need? Should I eat a power bar in the car on the drive there so that I'm not hungry while I watch my kid eat pizza?"
Where we live, it's typical for parents to drop their kids off at a party, even preschoolers. When I have parties for my toddler or preschooler, I make sure to invite entire families - parents and all siblings. And I mention that there will be meals and cake for adults and kids. Then, there's no question about if they're leaving the kids, if they're welcome to stay, if they will go hungry or be expected to pay their own way.
My DD is 5. Recently, she was invited to a birthday party. For the location, only an address was provided. It looked like a house address so I wrongly assumed that was the case. Every other invitation we've had before this has stated explicitly "At the park, address is..." or "At the indoor place with trampolines, address is..." not just an address with the parents' names on top.
I emailed the mother to see if we were supposed to leave the kids or stay with them. DD's name was the only one on the envelope so I imagined that meant that only she was invited. I didn't want to stay and seem like I was saying "I don't trust you" but I don't know the parents other than in passing, so I don't know them well enough to trust them. I figured if she would be offended or annoyed that I stayed at her house, I would just bring a book and park in front of the house, and tell DD if she needed anything, I was just outside and she could come get me. I wanted to ask the parents and get a feel for what they expected.
Here's part of what I wrote:"Also, do I drop DD off with you all and come back and pick her up or will parents be staying?"
She replied that parents were free to leave because they have the place all to themselves. Saying "the place" was when I realized that it might not be at their home. I had to search on the internet to see what was at that address. Google said it was a store that sold antiques. That didn't make sense. I did more digging. That antique store went out of business a couple of years ago. So it was a building in a commercial district but I didn't know what it was. I had to dig more on the internet and found that it's a new kids' party place with bounce houses. There will likely be tons of other strangers there. No way am I leaving my 5-year-old alone in a place like that.