The Shulhan Aruch, in discussing the laws of respecting parents (Yoreh De’a 240:2), mentions that it is forbidden to contradict one’s parent’s statement in the parent’s presence. If one’s parent expresses an opinion, it is forbidden to tell the parent that the opinion is wrong. In fact, one may not even support his parent’s viewpoint by saying, “I think he’s right,” because it is disrespectful to indicate that the parent needs the child’s backing. Certainly, the child may not explicitly state that he thinks the other view is correct.
As for situations where the parent is not present, the Poskim instruct that one may express disagreement, but in an indirect or otherwise respectful manner. One example given is an instance where the Tur (Rabbenu Yaakov Ba’al Ha’turim, 1269-1343) disputes a Halachic ruling of his father, the Rosh (Rabbenu Asher Ben Yehiel, 1250-1327). But instead of explicitly stating that he did not accept his father’s position, the Tur instead cited the ruling of the Ba’al Ha’terumot (Rav Baruch Ben Yishak of Worms, 1140-1211) – which was the same ruling as the Rosh – and then disputed his ruling. This way, the Tur disputed his father indirectly, rather than explicitly saying that he disagreed with his father’s position. Likewise, in any circumstance where one disagrees with his father, he may express his disagreement when his father is not present, as long as he does so with the utmost respect and reverence.
The Pit’hei Teshuba, citing the Asmot Yosef (commentary to Masechet Kiddushin by Rabbi Yosef Ibn Ezra, Bulgaria, 1506-1589), notes that this Halacha does not apply to raising questions and posing challenges over the course of study. If a father and son are learning together, it is certainly permissible for the son to question what his father said as normally happens when two study partners learn together. This is part of the normal process of learning, and thus does not constitute an infringement upon the father’s honor.
Summary: One may not dispute his parent’s statement in his presence, and even when his parent is not present, disagreeing with something the parent said is allowed only if it is done respectfully. When a father and son learn together, however, it is permissible for the child to question and challenge what his father says as part of the usual learning process.