In ALL the parties we have attended... that were held at a venue like Chuck E Cheese... the Host of the party.... ALWAYS ALWAYS... stated IN the invitation.... that parents can stay or drop off. BUT.... the invitation ALSO stated, that the Host/the Husband/and other relatives, WILL BE THERE at the party, in order to SUPERVISE the children. AND attend to them, per bathroom needs etc. AND that, the parents are responsible for leaving their contact phone number WITH the Host of the party. Too.
OR if the parents stay.... the Host is only paying for lunch, for the kids who attend. And not other people that the parents may drag along with them to the party, uninvited.
AND the invitation states (in order to control head counts) that: "Your child and ONE parent, is invited to the party." If others attend, lunch is not provided.
And of course, the RSVP deadline is stated AND a phone number and e-mail address.
And no, a single parent/Host of a party like that, cannot be responsible for a WHOLE group of kids. By themselves.
AND, as an attendee of the party, since you are the Parent, you simply contact the Host of the party, and ASK about supervision at the party and IF there will be enough "Adults" there, to supervise all the kids. That is what I do.
I personally, only drop off my kids at parties:
1) if they are comfortable with that
2) IF I know the family VERY well and my kids know them VERY well.
3) And IF I know... that there will be other Adults there/of the Host... that is there to supervise the kids.