It would seem to me that they do. The mom is blood related to everyone but the dad. The dad is blood related to 3 of the kids, who are blood related to the mom and the other kid. It seems kind-of silly.
If this was purely about making sure that everyone living in the house is somehow related, I think the board would have ignored the fact that one of the kids is not related by flood to the man in this case. Their living arrangement is clearly within the spirit of the law.
The board's reaction leads me to believe that this is more of a case of the board members using their power to show their disapproval of the fact that these two are not married.
Certainly, people have the right to morally oppose unmarried cohabitation. But to use government power to keep two property owners from living in their house? That's just asinine.