What is the ruling on the shroud used for the deceased father who died in an accident, and is the son obliged to pay the value of the shroud from his private money if the shroud is impermissible?

There is no harm if the charitable organization or another entity donates the shroud for your father and bears the expenses of his transportation and burial. You are not required to pay for the shroud. However, if you spend something from your own money as charity for yourself or for your father, it is a good deed for which you will be rewarded.