Yes, it's possible. Since your father is receiving SSI benefits, that means he has very low income and few assets and probably qualifies for the state's Medicaid program. It means he might also qualify for a state-run program -- if your state has one -- that can provide some payment directly to your father for care at home. If you're the person primarily caring for him, this money can then go to you. The payments aren't great, but they can help.
This payment for home care would come from a program usually run through the state's Medicaid program. It's called Cash and Counseling or something similar (the names may be different in different states). If your father qualifies for home care under Medicaid, Medicaid would normally provide that care through a certified home care agency. But this special program would instead pay your father directly the same amount Medicaid would pay an aide from a home care agency. If your father qualifies, the money can then go to you. You and your father can also use some of the money to make home improvements for his safety or comfort, or to buy personal care items for him.
To learn more about these programs -- not all states have them -- go to the page on this site called How to Get Paid for Being Your Parent's Caregiver. You can help your father apply for Cash and Counseling or similar state cash assistance program, if your state has one, at a local Social Service or Medicaid office -- probably the same office that administers his SSI eligibility and payments. If you don't have contact information for the local Social Services or Medicaid office near you, contact the Eldercare Locator by phone toll-free at 800-677-1116. Or, online you can go to any search engine and type in "Medicaid" and the name of your state; that takes you to the state's main Medicaid Web site, which can provide you with the contact information for the office nearest you.