The invitation goes to the CHILD, not the family. Put drop off and pick up times - that ought to be clear. When parents arrive, say "Hello, nice to meet you" (or "nice to see you" if you know them), make a little small talk, and then say, "Well I need to attend to the party. See you at 3:30!"
If you have someone staying to help you, make it clear that this person is a chaperone and not just a hanger-on parent. If your party is at a place with a specific activity (play space, arcade, bowling alley, whatever), put on the invitation that you will have excellent supervision.
The siblings are not invited - don't put down that it's at the parents' expense. Your daughter is not inviting the siblings to her party. Stick to that. If you have a farewell/see you later speech with a big smile and act like of course these people are leaving, there will not be a problem.
If anyone asks you if the 3 year old can stay, just say that it's really a small party for her special 7 year old friends, and not designed for a big crowd or other kids. Say that you know their 7 year old child will have a great time. Stop there. Don't keep explaining why you can't afford this or you wish the others could come but the parents have to pay. I commend you for having a normal-sized party and not a full-class free-for-all! Too many people are breaking the bank over these parties.
Also make SURE that your daughter writes a thank you note for her gifts. If people have the whole class, the birthday child gets overwhelmed and resentful about 25 thank you notes. But writing 6 or 8 notes is doable and important. It teaches your daughter about manners and gratitude, and teachings the guests that their presents and presence were appreciated.