Rav Menashe Klein, the Ungvarer Rov (Mishneh Halachos 6:158), was asked by a young man if he was permitted to go to learn in a yeshiva against the wishes of his father. The Rov refused to answer his question because the questioner did not write to him the reason for his father’s objection.
In principle, says Rav Klein, this is an old question dating back to the Trumas HaDeshen and brought in Shulchan Aruch (YD 240:25). The halacha is that if a talmid wants to go learn by a rebbi, even far away from home, and his parents are nervous about the physical dangers, he may go against their wishes.
However, there are certain objections the father can raise that are valid and then the son is not permitted to go learn there. The Gemara in Kesubos (71a) says that if a husband forbids his wife from going to simchos or the visit aveilim, she can demand a divorce and receive her full kesuba. However, if he claims that there are people there who behave immodestly, if there is strong basis for his claim his objection is justifiable.
Similarly, if the father feels that the rebbi lacks yiras Shaomayim or that the other talmidim may be a bad influence, then his objection stands, but only if there is validity to his claim.