Do I pay taxes for home health care workers?

Yes...you reall y need to pay taxes, and do withholding.

IRS says that home help is ALWAYS an employee

If you fail to withhold taxes and pay the employer share of Social Security, etc....when you are caught (not if...when) you will be solely liable to pay all those taxes out of your own pocket..and the fines and penalities. Really expensive!

I just hired a payroll service to handle it for my Dad and they do the filing, etc. I get the accounting once a month..and they get a check to cover the quarterly payments to the IRS and State dept of revenue.

You need to have signed agreements, never hire anyone without them, never pay more than the contracted wage. If you "give them extra money" Medicaid will see it as gifting and they STOP your benefits.