You either follow the contract, or you don't and risk being terminated and sued.
The contract is clear that you're to have no communications with the families other than from the place of business, and related only to business, and that you can't even write down a name or phone number in your address book.
The question is, "is this reasonable?". As you observe, the world of dance instruction is small. I also know that it's full of big egos, cut throat business practises, and that it only takes a couple of students and families to make or break one's reputation and standing in the community. Non competition is taken very seriously in the "fine arts" and "performance arts". I've seen equally onerous contracts in the world of horsie dressing and showing, and even in high end hair/nail services.
Naturally, these are the fields where the students/customers are far more loyal to their favourite teacher or stylist than they are to the business name or business owner. Hence these contracts.
You've signed it, and it comes to this: you abide by the terms or you risk being dismissed and / or sued. The question of course is, would they be successful? That depends on the facts. If you socialized, and got half of your employer's students to follow you to your own studio, that's a serious breach. If you run into a parent at Starbucks and they buy your coffee and ask you to join them at the table, that's far less serious of course. And there's everything in between.
If your primary goal here is to socialize in the dance community, ask some other instructors what they think or what they do in your city. If you volunteered to teach dance at a local community center, and met people that way (as well as built up contacts in the community of course) I doubt that your employer could say much about that. You're not charging for your time, thus you aren't in competition.
I know what you want me to say: that this contract is overly restrictive and that your employer shouldn't have any say over what you do in your social time. You're right, the employer shouldn't. And unless or until you go to work for the competition, or open your own studio, or are believed to be teach the students of your "friend families" differently from the rest, you shouldn't have a problem. The contact is of course meant to stop you from poaching students, not meant to act as a restraining order on your social life.
Does that discussion help? What are you ultimately trying to accomplish here?