No, only the parent that pays more than half the cost of maintaining the home can claim the head of household status.
Please see below for more information.
Which parent can
claim the dependent on their tax return if we live together but aren't married?
If you're living
with the other parent and aren't married and don't have an agreement, the
parent who has the highest adjusted income for the year has the right to claim
the child on their tax return. However, you're allowed to have an agreement
between the two of you about who will actually claim the child. In the case of
an agreement, you won't have to fill out Form 8332. Keep in mind, the parent
that claims the child as a dependent is the only one allowed to claim all the
child-related credits such as the Earned Income Credit and Child and Dependent
Care Credit. The other parent should not list the child on their tax return.
What about head of household filing status?
The parent claiming
the child is also the only one who can claim the head of household filing
status. This means the parent who claims the child would have paid more than
half of the expense of maintaining the household.
If you have two
children and each of you claims one, you still cannot both file as head of
household because only one of you can be paying more than half the cost of
maintaining the home. Also, if you split all expenses down the middle, then
neither of you can claim head of household.
