What is the ruling of Islamic law on selling a human organ (kidney) to pay off debts and secure a decent life for the family, especially given the difficult circumstances the questioner is experiencing, and is it permissible to undertake this act without the family's knowledge?

It is not permissible to sell human organs because the human body is not property to be offered for sale. A resolution has been issued by the Islamic Fiqh Academy prohibiting this. Furthermore, the parents' refusal of such an act confirms its prohibition, as it involves disobedience and harm to them. The questioner should seek Allah's help and search for lawful means to pay off his debts.