How can one balance obedience to parents in starting and completing the fast according to their view, and not contradicting other people, while avoiding the parents' anger or their severance of ties?

Obeying one's parents in matters that do not involve disobedience to Allah is one of the greatest acts of righteousness. However, it is not permissible to obey them in acts of disobedience to Allah, as Allah the Almighty says: (And cooperate in righteousness and piety, but do not cooperate in sin and aggression) [Al-Ma'idah 5:2]. It is also not permissible to differ from the people of the country regarding fasting and Eid, as the Prophet (peace be upon him) said: (The fast is the day you fast, and the Eid al-Fitr is the day you break your fast, and the Eid al-Adha is the day you offer your sacrifices). Disagreement is evil, so it is obligatory to be with the people of your country. Therefore, it is not permissible to follow your parents in their disagreement with the Muslim community regarding fasting and Eid. You must advise them kindly and seek help from scholars. It is not permissible to obey them in disobedience to Allah, even if they become displeased.