How to Reconcile Between Parents
Question

My father does not speak to my mother or interact with her; he does not speak to her at all or interact with her, and he does not return her greetings of salam. He has even been sleeping in a different bedroom for approximately a month and a half, up till now. 
Please note that my father is 58 years old, and he is a smoker. He has told me that he does not want her. My brother and I know that our mother is not lying to us, and that there is no reason for him to treat her like that. He does not even eat with us or sit with us. Is it haram for me to not speak to him? What must I do to put things right in the home?
Praise be to Allah.We appreciate your keenness to put things right at home and for asking about the way to reconcile between your parents. We ask Allah, may He be exalted, to make it easy to reconcile between them, and to grant you reward for that; we ask Him, may He be exalted, to guide your parents to that which He loves and is pleased with.
Virtues of reconciling between two disputing parties
Undoubtedly reconciling between two disputing parties is regarded as a great good deed in Islam. Allah, may He be exalted, has stated that this is one of the greatest of actions, as He, may He be exalted, says (interpretation of the meaning):
{No good is there in much of their private conversation, except for those who enjoin charity or that which is right or conciliation between people. And whoever does that seeking means to the approval of Allah - then We are going to give him a great reward.} [an-Nisa 4:114]
He, may He be blessed and exalted, also says regarding the one who refrains (from reconciling between people) because of an oath he swore:
{And do not make [your oath by] Allah an excuse against being righteous and fearing Allah and making peace among people. And Allah is Hearing and Knowing.} [al-Baqarah 2:224]
And He, may He be exalted, enjoined that in the verse (interpretation of the meaning):
{So fear Allah and amend that which is between you.} [al-Anfal 8:1].
If this is the case with regard to reconciling between disputing parties who are strangers and are not related, then reconciling between relatives is nobler and greater, and will bring a greater reward. Even greater and nobler than that is reconciling between spouses, because reconciling between them will create a peaceful atmosphere in the home, and will enable them to give a good upbringing to their children, because estrangement between them has a negative impact on their children.
Severing ties with parents
It is clearly wrong to sever your relationship with your father . On the one hand, it is haram, and on the other hand it will exacerbate the estrangement between your father and mother, and will add another dimension to the matter. What is required of you and your brother is to seek to put an end to this estrangement and dispute, and not exacerbate it by adding an extra dimension to it.
Tips on how to reconcile between parents
What we advise you to do is the following:

Offer supplication to Allah, may He be exalted, praying sincerely that He will help your parents to reconcile.

Supplication (du‘a) is the weapon of the believer, and you should seek out the best of times for it, namely the last third of the night. You should also seek to offer supplication in the best position, which is when prostrating.

Bring in two righteous and mature-thinking arbitrators, from your father’s family and from your mother’s family, to judge between them concerning that in which they differ. There is the hope that Allah will bring about reconciliation between your parents if you make a good choice of arbitrators. Allah, may He be exalted, says (interpretation of the meaning):

{And if you fear dissension between the two, send an arbitrator from his people and an arbitrator from her people. If they both desire reconciliation, Allah will cause it between them. Indeed, Allah is ever Knowing and Acquainted [with all things].} [an-Nisa 4:35]
Shaykh Tahir ibn ‘Ashur (may Allah have mercy on him) said:
“Regarding the words of Allah {If they both desire reconciliation}, what appears to be the case is that they refer to the arbitrators, because the context of the sentence refers to them. The Quran only mentions one condition, which is that they should want to bring about reconciliation, because this should be the aim of the guardians and arbitrators. Thus the duty of the two arbitrators is to examine the issue of dispute between the spouses, with the intention of bringing about reconciliation. If they manage to bring about reconciliation, all well and good, otherwise they may decide on separation, and Allah has promised them that He will help them to reach an agreement if they intend to do good. What is meant by causing it between them is helping them to reach the right and realistic conclusion, because if they both agree on a conclusion they will both be content with the conclusion they reached, in contrast to what would be the case if they differ.” (At-Tahrir wa’t-Tanwir, 5/47)

Remind your father of the virtue of overlooking and forgiving, and the impact of reconciling with your mother on the atmosphere in the home, and on his children, because a gentle reminder may be effective in narrowing the gap. You can remind him of the following verses:

{And to forego it is nearer to righteousness. And do not forget the graciousness between you.} [al-Baqarah 2:237]
{and who restrain anger and who pardon the people - and Allah loves the doers of good.} [Aal 'Imran 3:134]
{and let them pardon and overlook. Would you not like that Allah should forgive you?.} [an-Nur 24:22]

Remind him that it is haram to shun someone with no excuse, and it is haram not to greet your mother with salam.
There is nothing wrong with lying to both parties in order to bring about reconciliation. So you could tell your mother that your father loves her, and that he is very fond of her, and you can tell your father that your mother misses him and loves him, and wants the relationship to go back to being better than it was before.

It was narrated from Umm Kulthum bint ‘Uqbah ibn Abi Mu‘it that she heard the Messenger of Allah (blessings and peace of Allah be upon him) say: “He is not a liar who reconciles between people, conveying good things and saying what is good.” (Narrated by al-Bukhari, 2495)
For more information, please see the answers to questions no. 60316 and 112020 .
We ask Allah to reconcile between your parents and to guide them and enable them to do that which He loves and is pleased with.
And Allah knows best.