How can a young man combine worshipping Allah, as the Prophet (peace and blessings be upon him) did, with pleasing his parents, especially after his father accused him of disbelief due to an argument with his sister and his refusal to attend Friday prayer?

Your father's statement is serious. However, if he said it while not believing that what you did—quarreling with your sister—is part of the religion, then he does not exit the fold of Islam thereby. He might believe that the best way to address your sister's mistakes is through gentleness, kind words, and good counsel.