@itonewbie so correct me where I am wrong.  

The child is a qualifying child of both parents in this situation.  

For our discussion let's assume that the father did pay more than half of the support of the child.  It is not clear from what was posted but for this discussion let us use that as a fact.

Where in the Prop Reg or the Notice that you posted does it say that you have to claim the child as a dependent in order to claim Head of Household?

It doesn't, it speaks of having a qualifying child to be able to claim Head of Household.  He does have a qualifying child.