"No stag or hen parties" means the owner of that place wants, let's say, less boisterous tenants. It doesn't mean York is infested with boisterous people unencumbered by spouses. It's a big enough city that you and your five kids (non-boisterous no doubt) will find plenty to do without excessive exposure to stags or hens. If you're looking for a place big enough to accommodate all of you, you're looking at a pretty big place, which is why the owner is concerned about groups of adults who might do damage or upset the neighbors.
Your post says you want an Airbnb "for this summer." If you mean for the whole summer, you might have a better trip if you move around a little more. York is great but a whole summer there means missing a lot else in Britain. Or maybe you mean "for a week this summer" or some such, in which case it's a great choice.