The first question is: Is Aid and Assistance the same as Medicaid. It seems not. A&A is available to Vets with a VA pension.
Notice 2014-7:
This notice provides that certain payments received by an individual care provider under a state Medicaid Home and Community-Based Services Waiver (Medicaid waiver) program, described in this notice, are difficulty of care payments excludable under § 131 of the Internal Revenue Code.
I can't speak to how the funds from A&A are allowed to be used. You can charge your parent rent for a portion of the home. You can charge him for the care you give him. Rent is entered on Schedule E. You can take depreciation for the improvement to the basement.
If he pays you for care, this will be entered on Schedule C as self-employment income.
He can give you a gift of up to $15,000 per year without filing a Gift Tax Return.
All of these are possibilites, but again, this relates to income tax. You will have to determine which of these is possible with the A&A guidelines.