Do you have to continue paying the mortgage if a parent passes away?

If the payments don't get made, the lender will foreclose. There are many questions embedded in this one; you probably should start to administer your father's estate; however, there may only be liability, and no benefit to doing it. If there's a house, then potentially there's a lot of money on the table.

Maybe, maybe not depending on facts you need to discuss with a lawyer (and yes, get one ASAP).

Unless you signed the mortgage note, you are not liable. If you believe that the value of the property exceeds the mortgage and you are the sole heir that will inherit the property, you may want to work with the mortgage company to delay the foreclosure action that will ensue if payments are not made. If you have siblings that would also inherit the property, you will want to have a written agreement with the others as to being repaid for any funds you put up to forestall the mortgage foreclosure. Otherwise, the mortgage will likely be pushed into foreclosure when the payments fall 90 days overdue.

You have no legal obligation to make payments on your dad's mortgage. If you don't the bank will foreclose on the house. If the house is worth more than the mortgage (or if you are living there and the mortgage payment is less than rent you would pay for an equivalent rental), make the payments. If not, don't.

The mortgage follows the property, so if you inherit you will have to pay the mortgage.