If your parent had any more than $4300 of income in 2021 not counting Social Security, you cannot claim them. The unemployment "counts."
Your parent can file their own tax return to seek a refund of tax withheld. However, unemployment is taxable on a federal return for 2021 and maybe on their state return too.
It's possible. You would need to prove that you supply more than half their support.
"The first requirement for filing as head of household is that you must have paid for more than half of the expenses involved in maintaining a qualifying household during the tax year. This means that you must have paid more than half of the total household bills, including rent or mortgage, utility bills, insurance, property taxes, groceries, repairs and other common household expenses. In the case of a parent, your parent doesn’t need to live with you, but you must maintain at least half the cost of their living arrangements.
If you receive financial assistance toward your household expenses from a parent or other individual, you can still qualify to file as head of household as long as you are paying for more than 50 percent of the bills with your own earnings, savings, or capital."
How would this change if they received more than $4,300 of unemployment help from the state agency? Basically had them keep as much as possible from the unemployment benefits they're receiving throughout 2021 while I continue paying for bills. Do I still qualify as HOH?
each parent stands on their own. you can't claim the one having gross income of over $4300. if both had over $4300 in gross income you can't claim either.
in addition to gross income to claim a parent you need to have supplied more than 1/2 of their support.