Is money deposited in a bank from car accident compensation permissible (halal) or forbidden (haram)? And what should be done with it if it is forbidden? And does the ruling apply to the money of the mother and father?

It is permissible to take the aforementioned compensation and dispose of it. The ruling regarding its permissibility or impermissibility is the exclusive domain of the Sharia courts. It is permissible to deposit money in Islamic banks. As for usurious banks, it is permissible to deposit money in them if it is the only means of preserving it, provided that the money is placed in a current account that does not accrue usurious interest, and any interest earned is spent on general Muslim welfare.