Is one obligated to make up (the days) or offer expiation for breaking the fast in Ramadan out of fear that his parents would discover his conversion to Islam, and the consequent cessation of his study expenses?

A servant's Islam is accepted if he utters the two testimonies (Shahadatayn), and he is not required to declare it publicly if he fears severe consequences, similar to the believer from Pharaoh's family who concealed his faith. If you break your fast in Ramadan due to fearing severe consequences, you are excused and must make up that day when you are able, and no expiation (kaffarah) is required of you.