Whomever the probate court appoints.
The estate.
The contents of a Will are your parents' personal and legal business. It is none of a child's business unless the parent shows you the Will, includes you in estate/will planning, or you are named the Executor/Executrix of the Will (this is not a minor child). When the parents die, the Executor/Executrix can contact the parents' Attorney to get a copy of the Will, if it's location is unknown at home.
The court will appoint an executor.
An executor handles the estate of a decedent who died with a will. An administrator handles the estate of a decedent who die without a will. The terms are different because an executor is executing the decedent's directions as stated in the will. The administrator is simply handling the estate according to general laws.
The sister in law's status as a tenant in common does not give her any special advantage to be appointed as executor. The daughter should continue with her petition for appointment as her parents' heir. As tenants in common, the parents' interest in the property will not pass to the sister in law.
Typically the estate is responsible for clearing up any liabilities. The affairs of the estate are usually handled by the executor of the estate.
Where I live the executor of your estate will have to file your Federal taxes after you die.
A testator chooses an executor to take charge of and settle their estate after they die. The executor must file the will in probate and must be appointed by the probate court. The executor has no legal power until they've been appointed by the court. The court generally will appoint the executor that was named in the will unless someone files a legitimate objection. The executor has nothing to do with your husband's right to inherit from your estate. If you die owning property in your own name, and you don't live in a community property state, the property will be distributed according to your will, or, if you have no will, it will be distributed according to the state laws of intestacy. You can check the laws of your state at the related question link provided below.
The executor must make the payments from any assets of the deceased Estate until the Estate is settled.
The estate is
The debts of the parents are paid by the parent's estate, not their children.
The specifics will depend on the executor of the estate. If when the parent's die there are not adequate assets to settle the debts, the loan will probably have to be repaid. If there is enough to cover the debts, the loan can be subtracted from the bequest.