It probably makes sense for your parents to transfer the house to you. Usually such a transfer can cause up to five years of ineligibility for Medicaid coverage. However, it sounds like you would qualify for the so-called “caretaker child” exception to Medicaid's rules governing the transfer of the home.
In most states, to qualify for this exception you need to get a letter from a doctor saying that for at least the past two years were it not for the care you provided to your parents, they would have had to move to a nursing home. That said, since every state interprets these rules differently I strongly recommend that you consult with an Alabama elder law attorney.
Read more on Medicaid's asset transfer rules.