My parents have both passed away within the past 3 months. We have continued to pay the mortgage so it remains current. What can we expect the mortgage company to do once they are notified that both people who took out the loan have passed away?

The mortgage will be taken over by the estate. They will normally have the heirs assume the mortgage, refinance a new mortgage, or sell the property and pay the mortgage off. The person who is handleing the estate, should notify them and the heirs should decide what they want to do with the property.