If kids whom I (and my child) didn't know were in my yard, I would immediately call them over, get out my cell phone and say they needed to call their parents, tell their parents where they are, and then I would get on the line to let the parents know that Sally and Jimmy are in my yard because they are friends of my neighbor kid Billy, whom I know and whose family I know, but I don't know you (the parents) or the kids. I would then say, (actually I personally wouldn't but you could say) that I'm fine with the kids being there and playing but (1) I have to have the parents' name and contact numbers for right now, (2) the kids need to be picked up by a parent at time X tonight; and (3) I want the parents to come by and meet me before their kids come over again.
If you really do like having all these kids around, that's great, but frankly I think you need to consider liability -- as someone else noted, you are liable if a child is hurt on your property -- and the fact that a stranger may not be as understanding as an acquaintance if her child is hurt at your house; your neighbors and their kids know you and have for a while, but some stranger whose child gets hurt at your place may not hesitate to go after you if something happens.
I know -- it's not great to think that way; we want to assume the best of people; and so on. But honestly, 27 kids, many of them unknown to you, and you the only adult present? That's not an adult-to-kid ratio that would be acceptable in many schools and camps, depending on the kids' ages.
And doesn't this casual arrangement to come over and play for hours and hours and hours (including what sounds like meal hours) cramp your family's style when it comes to eating when you want, going out on a errand, just going out at all? What if you need to leave and the yard is full of kids and you can't reach every parent (and how long will it take to make 15 to 27 phone calls?!) Do you usually shoo off the kids when you need to say, "I have to go out with my kids now, you all have to leave, see you later"? Do they actually depart or is there a chance they're going to just stay on your property unsupervised, which is asking for trouble?
The situation sounds pretty untenable to me. Even if you meet everyone's parents -- it seems you are being taken advantage of as free babysitting and the word has spread to families you don't even know.
The best defense for standing your ground, as you put it, may be to be off your ground entirely -- if you are not there and your kids are not there, there's nothing for the kids to come to visit. I'd be very busy all summer long. And when home, if I were not up for having kids over, I'd tell them, "Sorry, we're busy right now and this isn't a play day at our house. Time to call your mom and dad to come walk you home." With a smile but firmly. You really do have a potentially tough summer on your hands if you don't lay down the law like this right now.