How can a new Muslim perform prayers in their Christian parents' home while maintaining the secrecy of their Islam?

Your duty is to practice the rituals of Islam, such as prayer. If you are coerced into not openly performing your prayer, then pray according to your circumstances. As long as you fear for yourself, you have a concession to combine the prayers (Dhuhr with Asr, and Maghrib with Isha), either by performing them earlier or delaying them. If you are unable to do that, then pray the prayer of one in fear (salat al-kha'if), performing what you are able to of the prayer's conditions and pillars, and what you are unable to do is waived from you. This concession applies in cases of compelling fear that would result in significant harm, and not merely shyness towards family members.