What is the ruling on working in an internet café whose devices are used for browsing pornographic websites, and is it obligatory to leave this job if the father does not agree to it?

If you are able to prevent customers from accessing forbidden websites, then you must do so. Otherwise, you must abandon this work, because warding off evil takes precedence over acquiring benefits. It is not permissible for you to assist patrons of the shop in using your devices for what Allah has forbidden, as this constitutes cooperation in sin and aggression. It is not permissible to obey your father in this matter, for there is no obedience to a creature in disobedience to the Creator. We advise your father, if he is able, to exempt you from working outside the home, to avoid being alone and mixing with non-mahram (definition) men.