Coming to an older person's home and doing an assessment of potential problems there makes the most sense - and the likelihood of having such an in-home assessment funded by
the state varies across the nation. You might try contacting their primary care physician, and asking if he will write an order for physical therapy evaluation by a medical home care agency. This should be covered by Medicare. Hopefully, your mother will qualify for several therapy visits aimed at strengthening the muscles needed for a safe transfer.
Part of the information you need in order to be an effective caregiver is their financial assets. You can explain to them that you are willing to spend you time identifying sources of assistance for them but you need to know what programs they will qualify for financially. If your father was in the military during wartime, there are VA sources. There are Medicaid resources, too. Both have financial limits.
They are fortunate to have a caring daughter who obviously wants to help.