What is the ruling on preventing one's sister from watching channels that contain music after she used to watch them under your guidance? And what is the ruling on the mother's statement that a cooking channel with music is permissible because the good in it outweighs the evil? And what is obligatory to do if the father orders these channels to be reinstated?

Obedience to parents is obligatory in what is good, especially the mother. If the parents command to open channels that do not violate the Sharia (definition), then obedience is obligatory, and if the sister requests it, the brother has no right to prevent her. As for channels that contain evil, it is not permissible to open them, as this constitutes aiding in sin, and there is no obedience to parents in disobedience to Allah. If the son is not the one in authority in the house, he is not permitted to control channels locked with a password and prevent his family from benefiting from them; rather, disapproval and verbal advice suffice.