Is it permissible to take out a loan from a bank and engage in the stock market to help one's father pay off his debts, given difficult circumstances and the need to get married, or is this forbidden, and what is the solution in this case?

It is not permissible to take interest-based loans except in cases of dire necessity. Paying off due debts is not a necessity that permits this, because Allah commanded to give the one in hardship respite, and charity to him is better for him. Usury is a declaration of war against Allah. Furthermore, engaging in the stock market for trading shares and bonds is often not free from prohibitions due to its non-compliance with Sharia regulations, and some joint-stock companies engage in prohibited activities or deposit their money in usurious banks. If there is no way to pay off your father's debt, then you must be patient.