Most of our birthday parties (both given and attending) were held within the bounds of classmates from the Montessori preschool our kids attended. As such, almost all the parents not only knew each other but stayed at the party. We typically always provided both kid and adult foods and beverages (sodas and some beer), and we often had as many as 25 children under five with no fights or arguing. I love Montessori techniques!
A few times we were invited to other birthday parties, more like the ones you are describing. While some parents simply dropped the kids off, I always asked the mom if I could stay and give her a hand, and they always said yes.
The one exception was a party given by a British mom. She allowed several of us parents to sit in the formal living room, with crustless cucumber and butter sandwiches and a big pot of tea, and proceeded to hold an all-girl party like one out of a book. Every activity was choreographed completely, she planned every last detail, and she needed no outside help. The kids had no choice but to toe the line, and the parents felt a bit out of their league.
When my kids were little (and much to dismay, still now that they are teenagers) I wanted to be certain that they were both safe and not causing a problem.