Most health insurance will cover the cost of a family member caring for the person in their home but but excludes spouses and legal guardians.
But if you're any other type of relative to the person you want to care for or you're caring for the person then usually yes you can get paid by health insurance companies to take care of a family member.
You usually must not live in the same house though to be paid by the health insurance to take care of the family member.
My sister takes care of her uncle in his home and gets paid by his health insurance so yes it's possible.