It is highly unlikely that you qualify for Head of Household (HOH) in the circumstance you describe.
In order to claim a parent, that does not live with you, as you qualifying person for HOH, that parent must be claimed as your dependent.
The fact that you paid for more than half the cost to run your parents' home is not relevant to you being able to file as HOH. But, it does help for the support rule for being able to claim a dependent.
You can not claim a person as a dependent, if that person files a joint return with their spouse UNLESS they are only filing to get a refund of withholding.
