Firstly: 

The oath that one swears and resolves to do must be fulfilled by doing what one swore to do or refraining from what one swore to refrain from. If he does not do that, then he must offer kafaarat yameen (expiation for breaching an oath). Expiation for breaking an oath has been discussed in detail in the answer to question number 45676. 

Secondly: 

If fulfilling an oath conflicts with honouring one's parents, then precedence must be given to honouring one's parents, and one must break the oath and offer expiation, so long as there is no sin involved in doing so. 

Muslim (1650) narrated from Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever swears an oath then sees that something else is better than it, let him do that and offer expiation for his oath.” 

Al-Nawawi said:  

In these hadeeths there is evidence that if a person swears to do something or not to do something, and breaking the oath is better than fulfilling the oath, it is mustahabb for him to break it but he must offer expiation. This is agreed upon. End quote. 

It says in Fataawa al-Lajnah al-Daa’imah, 23/124: 

If you swear by Allah not to take something in the future, then this is regarded as an oath that one must adhere to if adhering to it is an act of obedience to Allah. But if adhering to it is an act of disobedience to Allah and His Messenger, or you think that something else is better than it, then you may break your oath by doing something that is contrary to what you swore to do. End quote. 

Fataawa al-Lajnah al-Daa’imah, 13/124 

See also the answer to question number 9985.