Praise be to Allah.
If there is a necessary reason why this student should stay (with his parents), then it is better for him to do so. However, he can stay with them whilst also seeking knowledge, because honouring one’s parents takes precedence over jihad for the sake of Allaah, and knowledge is part of jihad. Therefore honouring his parents takes precedence over seeking knowledge if they need him.
But if they do not need him, and he will be able to seek more knowledge by traveling, then there is nothing wrong with him travelling to seek knowledge. But at the same time he should not forget the rights of his parents, so he should come back and explain to them when he comes back why he left. But if he knows that his parents dislike the pursuit Islamic knowledge, then in this case he should not obey them and he does not have to seek their permission to travel, because what is making them refuse is their dislike of Islamic knowledge.