The Gemara in Masechet Pesachim (51) mentions the prohibition against going to the bathhouse with one's father, father-in-law, stepfather or brother-in-law (sister's husband). The reason for this prohibition is that seeing these relatives unclothed might lead to improper thoughts, as they are associated with a person's roots and close family. The Gemara also mentions that one should not go to the bathhouse with his Rabbi, as this would be disrespectful. Some communities, as the Gemara discusses, had the practice of forbidding going to the bathhouse with one's brother, as well, but the Gemara mentions that strictly speaking this is permissible. Only communities that have adopted such a practice must abide by this stringency.
These Halachot are codified by the Rambam and by the Rama in Even Ha'ezer (23). The Rama there mentions that one may not enter a bathhouse with one's brother, either, but the Gaon of Vilna claimed that this must be a textual error, for as noted earlier, the Gemara clearly permits this except in communities that had the practice to forbid doing so. In a different context (Yoreh Dei'a 242), the Rama discusses the prohibition against going to the bathhouse with one's Rabbi, and he adds that if a person was in the bathhouse and his Rabbi entered, he does not have to leave the bathhouse. Halacha forbids only going to the bathhouse together with one's Rabbi, or entering if one's Rabbi is already there.
This prohibition applies as well to the Mikveh, where men see each other unclothed. It is therefore improper for a father to bring his son with him to the Mikveh.
Rabbi Moshe Feinstein (Russia-New York, 1895-1986), in one of his responsa, discusses the question of whether a person may perform the Tahara (preparation of a body for burial) for his father or father-in-law. Since the body is unclothed during this procedure, perhaps the deceased's son and son-in-law should not participate. Rabbi Feinstein does not reach a definitive conclusion on the issue, and he writes that it depends on the local custom. A person facing such a question should consult the local rabbinic authorities for guidance concerning that community's practice. If one cannot ascertain the local custom, Rabbi Feinstein writes, he should be stringent and not participate in the Tahara for his father or father-in-law. He further notes that some communities have the practice – which is based on the teachings of Kabbala – that a person does not go to his parent's grave at the funeral. Adherents of this practice, Rabbi Feinstein rules, should also refrain from performing a Tahara on one's father.
Rabbi Feinstein adds that this prohibition does not apply to grandparents; one may go to the bathhouse with his grandfather, and may thus also participate in his grandfather's Tahara.
Summary: One may not go to the bathhouse or Mikveh with his father, father-in-law, stepfather, sister's husband, or Rabbi. Whether or not one may perform a Tahara for his father or father-in-law depends on local custom.
