Is the described situation, which includes the cramped living conditions and their health and psychological impact on the questioner and her family, along with the inability to cut off support to her parents, considered an extreme necessity that permits taking an interest-based loan to build a house in the absence of an Islamic alternative?

Borrowing with interest is not permissible except out of dire necessity, which is likely to lead one to ruin, or cause unbearable hardship, or prevent one from achieving the minimum standard of living for the poor. We do not see an apparent necessity in the questioner's case; because she can rent a house that meets the need, and the husband is responsible for providing housing for his wife and child.