The simple answer is probably not, as most Mortgage documents and Mortgage NOTES have a "Due on Sale" clause.
A certified financial planner and an "elder care" Attorney experienced in Real Estate and Trust Laws in your state, may be able to fashion an irrevocable Living Trust for the property, which could possibly name your husband as survivor beneficiary, or fashion some other kind of structured trust that may be able to be protected from catastrophic medicade spend-down provisions, but generally there are waiting periods before one can be "in the clear" and the laws vary from state to state.
Unless the "mortgage" is an older one (pre-1988) it is highly UNLIKELY that there are "simple assumption" provisions, and that the mortgage note itself, and the recorded mortgage document as well may repeat the "Due on Sale" language. If so, unless a Formal Assumption is agreed to by the financial institution, AND they accept your credit worthy-ness, AND the appraisal is good, AND the title insurance is updated, no. By the time one endeavors for a Formal Assumption, one generally has already spent the same, if not MORE to maintain an older mortgage. Also..unless the interest rate and provisions are such that make this better for the institution to keep (higher than market rates versus your credit worthiness)..they wouldn't agree to it anyway. However the Due On Sale clause most-likely contained in Dad's original Loan documents would preclude this possibility.
IF MOM was on the original documents and is now deceased, it also matters what kind of "tenancy" status they held. If "joint tenants in common with rights of survivorship" then it could be possible to add parties via quit claim deed after her death certificate was filed ON the property an INTEREST, i.e. Dad to DAD plus son, but that can invoke the DUE ON SALE clause in some states.
A carefully structured trust for DAD may work in your state without invoking the due on sale, but it would remain in the trust until Dad died, and you would have no titled interest in the property, only upon his death as a beneficiary of the trust and again the mortgage would be due and payable upon his death.
Also keep in mind that your paying that mortgage is essentially a GIFT (gift tax kicks in from son at $10,000 I believe) to DAD, unless its RENTAL INCOME to DAD, both of which are taxable, and could pre-clude his ability to get financial help like medicaid. furthermore, if he is RENTING his home, it can be invoked for spenddown purposes, if he lives in it, it cannot.