Praise be to Allah.
If it is the obligatory Hajj then it is not essential to have their approval or permission. Rather if they try to stop him going to Hajj and it is the obligatory Hajj, then he has to go for Hajj and not obey them, because Allaah says (interpretation of the meaning):
“But if they (both) strive with you to make you join in worship with Me others that of which you have no knowledge, then obey them not; but behave with them in the world kindly”
And the Prophet (peace and blessings of Allaah be upon him) said: “There is no obedience to any created being if that involves disobedience towards the Creator.”
But if it is a naafil Hajj, then he should look at what is better. If his father and mother cannot do without him and cannot bear his absence, then it is better for him to stay with them, because a man asked the Prophet (peace and blessings of Allaah be upon him) for permission to go for jihad and he said to him: “Are your parents alive?” He said: “Yes.” He said: “Then your jihad is with them.” So in the case of the obligatory Hajj they should not be obeyed, but if it is a naafil Hajj, he should see what is better. End quote.