Where I live it seems to run that for "friend" parties (as opposed to family and close family friend parties) that the parent usually stay until the kid is in grade 1, and then the expectation is generally that the child is dropped off. Even with quite helicopterish parents in the 80's, my folks never stayed at McD's or such once I hit grade school. It was expected that the birthday host either asked the parents to stay (on the invite) or that they would arrange adequate supervision themselves.
I was a little surprised to have a couple of the 3 and 4 year olds left at my sons birthday last year, because in my mind strangers or relative strangers shouldn't have to help a child with toileting, and a 3 year old often needs help in the bathroom. The average healthy 7 year old won't, and they are already, in most cases, used to going to school where the "supervision" may only be one adult for 25 or so kids. I would have no problem leaving my seven year old at a play center for a party if he knew his home phone number, how to approach a responsible adult for help, who to go to if he was lost, and how to go to the bathroom independently. I could do all that at 7 and had no problems going to friends parties at homes and public places- I would have been the scut of much playground teasing if my mom had stayed with me!