How can a young man deal with his parents' opposition to his religious commitment, their demand that he not perform some acts of worship, and their accusation that he is lying, which led to him raising his voice at his father, and his feeling of difficulty in adhering to his faith under these circumstances?

Your duty is to be gentle with your family, treat them well, endure their harm patiently, and supplicate for their guidance. You must also repent to Allah for your bad conduct and raising your voice at your father, for that is a form of disobedience to parents. Your kindness towards them is a sign of your uprightness.