All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
The obligatory fasting, such as fasting as an expiation for an oath – when one is unable to feed poor people, or clothe them, or free a slave – it is not required to ask for the permission of the parents.
The person who broke the oath must fast even if his parents did not permit him, and he should inform them that this fasting is an expiation for an oath, so that they will be pleased with that.
Al-Bahooti from the Hanbali School of jurisprudence, said in Kash-shaaf Al-Qinaa’ ‘an Matn al-Iqnaa’ (2/385):
"The parents are not allowed to prevent their son from the obligatory Hajj, the Hajj of Vow, nor to tell him to come out of the state of Ihraam for Hajj, and it is not permissible for the son to obey them in this regard; i.e. in leaving an obligatory Hajj, or to leave the state of Ihraam (for Hajj); the same thing applies to every obligation, such as the congregational prayer, the Friday prayer, and traveling for an obligatory knowledge, because it is an individual obligation, so the parents’ permission in not considered, like in the case of the prayer." [End of quote]
Allah Knows best.