That. X being correlated with Y can mean that Z is causing both. "Fertility" and "living with parents" are each highly dependent variables, dependent on all sorts of things.
Let's remember "natural experiments." Aren't there any?
A country that suddenly deregulated housing? Where there were suddenly major tax incentives for having kids (looking at you, Hungary)? Two countries that are exactly the same except that one has dramatically different housing costs, or fertility?
Without some data like that, I'm not buying any causation arguments.