Is taking out a loan to pay off a father's debts a sin, especially since the borrower is the family's sole provider?

If you took an interest-bearing loan to avert a definite harm from your father, such as his imprisonment for a period that would harm him, and you found no other means to avert this harm, then there is no blame upon you. This is because it falls under the category of necessities that permit forbidden acts, in accordance with the Almighty's saying: "And He has explained in detail to you what He has forbidden you, except what you are compelled to [eat or use]" (Al-An'am: 119). It does not matter that the necessity pertains to your father, for your father's imprisonment is as distressing and harmful to a son as his own imprisonment, perhaps even more so. Some scholars have considered the threat of imprisoning a father as coercion against the son.