Should they join their father in celebrating the Gregorian New Year so that they can get some money from him for their mother who is poor?
Praise be to Allah
What we advise you to do when interacting with your father is to try hard to be kind to him and bring him back to the religion of Islam using a gentle approach, by treating him nicely and interacting with him kindly, being patient with his annoyance, and advising him in a gentle and kind manner, in the hope that he may pay heed, especially since you do not have any other means, and you are poor and have no choice but to live with your father, so you have no option but to deal with him gently and try to soften his heart. However at the same time you must hate what your father is following of disbelief, misguidance and sin, and you must stay away from him when he is committing sin, and not sit with him or give him the impression that you approve of it under any circumstances.
As for your joining him in his celebration of the so-called New Year, this is something objectionable and is not permissible, because it is well-known that the Muslims do not have any festivals except Eid al-Fitr and Eid al-Adha, and the weekly “Eid” that is Jumu‘ah. Any celebration of any other festival is not allowed, and can only be one of two things: innovation (bid‘ah), if it is celebrated as a means of drawing closer to Allah, such as celebrating the Prophet’s birthday (Mawlid), or imitating the disbelievers, if it is celebrating by way of custom or tradition, not as an act of worship, because introducing innovated festivals is the practice of the People of the Book whom we are commanded to differ from. So how about if this celebration itself is one of their festivals? This has been discussed previously in fatwa no. 145950.
What you have mentioned about needing to take money from your father to send to your mother, and that this is not possible except by joining him in this reprehensible celebration, does not justify your taking part in this grave evil, especially since your father is not obliged to spend on your mother’s maintenance because his relationship with her ended when he divorced her. Rather her maintenance is due from her children, so long as she is in need. Rather than taking part in this evil, it is better for you to look for a job that is compatible with your studies, and then spend on your mother from your salary, and thus become financially independent of your father.
And Allah knows best.