From Ben Olam Haba (a talmid chacham in Lakewood):
The Talmud (Pesachim 51a) teaches us that a son may not see his father undressed, and thus should not go to a Mikvah at the same time. If, however, they are in a bathhouse or swimming pool where bathing trunks are worn, there is no prohibition. (See Rama Yoreh Deah Siman 242:16 and Even HaEzer Siman 23:6)
I would say that this prohibition begins probably from the age of Chinuch when a child can grasp things of this nature, as Rashi explains the reason for the prohibition that a son seeing his father like this will come to have bad thoughts.
The Aruch HaShulchan (Even HaEzer Siman 23:8)wonders about the custom which is against halacha and tries to say that perhaps the Isur is only being in the actual bath/Mikvah at the same time, but not being in the same room. However, he himself says that based on the Rishonim, this cannot be a good heter. (also Rashi to Pesachim 51a explains the reason as the son may see his father and it will bring him to unclean thoughts , and that applies to them being in the same room naked, regardless if they actually enter the same bath).
He also writes there that it doesn’t apply to a young child who doesnt yet have unclean thoughts.
At what age this begins can vary from child to child and also even if relying on this heter, it has to be a crowded Mikvah, and not just a Mikvah where there are one or two other people there besides the father and son.