I don’t care how healthy your family is: you should let your kid have birthday cake. I’m not saying you can’t restrict your kid’s diet at all, especially if they’re going to a different birthday party every weekend. But outright banning them from consuming sugar during any occasion is wild. They should at least be able to have cake at their birthday party, even if they aren’t allowed to have it at everyone else’s. 

If you want to draw a line in the sand with your family, you must be clear about your boundaries. If you tell a 5-year-old, «You aren’t allowed to have cake,» and send him to a birthday party without informing the parents he’s not allowed to have cake, newsflash, he’s going to eat the cake. Parents aren’t mind readers, and it’s not a crazy thing to assume that if you send your kid to a birthday party without further instruction, the hosts will assume that the birthday party food is suitable.