It never would have occurred to me to stay over at an at-home birthday party unless parents were specifically requested to stay. If it was an activity party (i.e. McDonalds or the zoo) that's different; I would ask whether parents were included or not. If not, I probably would stick close to the venue (same mall or parking lot or just wander around on my own with the other kiddo).
If I were hosting, I certainly wouldn't be *offended* that parents wanted to stay but I might find it odd or surprising. Guess now I wouldn't be surprised, after reading this! But if a parent wanted to stay I guess I would ask them to help out in some way. Growing up I can remember maybe one or two moms helping out at parties, but they were always doing stuff, not just lounging around. I guess I see it as normal that way, if they're offering to, like, chaperone... serve cake or help with activities or lead games or something... instead of just looking out to make sure their own kiddo isn't doing something dangerous. Is that rude? To ask them to pitch in? (Politely I mean, not hand them an apron and demand they scrub dishes or something.)
But since we plan on homeschooling I doubt this issue will arise; unless we are friends with the parents the kids probably won't get mass party invites (but who knows what the future holds). And if I'm friends with the parents, I probably would just simply ask or assume we could stay. If we're not friends-friends... I probably wouldn't really want to stay unless DS/DD asked me to.