Praise be to Allah.
Undoubtedly obedience to parents or to one of them in that which does not involve disobedience towards Allah is one of the greatest righteous deeds and one of the best ways of drawing closer to Allah.
With regard to obeying them or one of them in that which involves disobedience towards Allah, that is not permissible.
Abu ‘Umar ibn ‘Abd al-Barr (may Allah have mercy on him) said:
The scholars are unanimously agreed that one who is instructed to commit evil is not obliged to obey. Allah, may He be glorified and exalted, says (interpretation of the meaning):
“Help you one another in Al‑Birr and At‑Taqwa (virtue, righteousness and piety); but do not help one another in sin and transgression”
End quote from at-Tamheed, 23/277
The scholars of the Standing Committee said:
Obedience to parents is prescribed in that which involves obedience towards Allah and that which is permissible. As for obeying them in that which involves disobedience towards Allah, that is not permissible.
End quote from Fataawa al-Lajnah ad-Daa’imah, 22/187
It is not permissible to differ from the local people with regard to the (beginning of the) fast and Eid, because of the general meaning of the words of the Prophet (blessings and peace of Allah be upon him): “Fast when you see it (the new moon) and break the fast when you see it.”
Narrated by al-Bukhaari, 1909; Muslim, 1081
And he (blessings and peace of Allah be upon him) said: “The fast (begins) on the day when you (i.e., the Muslim community) fast, the breaking of the fast is the day on which you break the fast, and (Eid) al-Adha is the day on which you offer your sacrifices.”
Narrated by at-Tirmidhi, 697; classed as saheeh by al-Albaani in Saheeh at-Tirmidhi
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
This was the practice of all the imams (leading scholars) of the Muslims.
End quote from Majmoo‘ al-Fataawa, 25/202
Imam Ahmad (may Allah have mercy on him) said: He should fast with the ruler and the main body (jamaa‘ah) of the Muslims, whether (the sky) is clear or cloudy.
Ahmad said: The hand of Allah is with the jamaa‘ah (main body of the Muslims).
End quote from Majmoo‘ al-Fataawa, 25/117
Ibn Baaz (may Allah have mercy on him) said:
Differing (from the jamaa‘ah) is bad. What you should do is be with the people of your country. When the Muslims in your country break the fast, then break the fast with them, and when they fast, then fast with them.
End quote from Majmoo‘ Fataawa Ibn Baaz, 15/100
See also the answer to question no. 12660
Moreover, their starting to fast one day after the people means that they will be breaking the fast on the first day of Ramadan, then they will be fasting on the day of the Muslims’ Eid. Both are haraam and are not permissible.
If we add to that the fact that they are going against the main body of the Muslims, it is even more haraam and the sin becomes greater. So it is not permissible to follow them in this reprehensible action.
What you must do is advise them in the best way and be gentle in guiding them. You can seek help in doing that by asking the scholars, so that they will realise that what they are doing is haraam according to the religion of Allah and is not permissible. It is also not permissible for you to follow them in that, even if they get angry and upset, because it is not permissible to obey them in that which involves disobedience towards Allah.
You should make the most of this good occasion, which is the presence of your brothers and their families, and their determination to differ from (your parents) in that regard, so that they may continue to do that, i.e., fast on the day when the people fast and break the fast when the people break the fast, and to differ from (your parents) in that regard and do so openly. This gathering of the family may serve as an opportunity to discuss this issue, whilst seeking the help of Allah and asking Him to guide them and open their hearts to the truth.
Then if they cut you off or shun you after that, there is no blame on you for their sin. Rather the sin is theirs, unless Allah guides them to the truth.
For more information, please see the answer to question no. 245
And Allah knows best.