"how does the IRS determine which of them can legally file as Head-of-Household"
One of the requirements for filing as head of household is that "You paid more than half the cost of keeping up a home for the year." Only the parent who paid more than half the cost of keeping up the home can file as head of household. The other parent obviously paid less than half. If they literally split all expenses exactly 50/50, to the penny, then neither one paid more than half the cost, so neither of them can file as head of household .
