All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
First, we ask Allaah to make a way out to your difficulties and reconcile your parents.
The stability of marital life is one of the most important objectives of Islam in legislating marriage so one has to endeavour in order to achieve this objective with all possible and permissible means. One of these means is spreading the spirit of understanding between the spouses, and that they should not suspect each other.
Therefore we advise you to strive in order to reconcile your parents. You have to be wise in doing so while seeking Allaah's help then the help of your trustworthy righteous relatives. Besides, you have to advise your mother in a good and soft manner.
Finally the following important matters should be noted:
Your mother should not be suspicious of her husband. In principle a Muslim is innocent until he is proven guilty. Moreover, suspicion leads her to spy on him and therefore destroy herself in sorrow because of that.
Her separation in bed from her husband and wearing Hijab in front of him is forbidden and she is considered disobedient to him; so this requires repentance.
When her husband lied to her, it was for an acceptable reason which was in order to keep good marital relationship with her, and this kind of lying is acceptable in religion. Imaam Muslim reported that Um Kalthoom Bint 'Uqbah said: "I have not heard the Prophet permitting telling lies in anything except in three situations: war, reconciling between people, and the talking of the husband to his wife and of the wife to her husband."
Similarly, your father should be reminded about the danger of mixing with women, and that it is not permissible for him to correspond with girls except in case of need while observing good morals. Moreover, he has to endeavour to find work elsewhere. He should also be reminded that Allaah has put the protection and maintenance of the wife in the hand of the husband. So failing to observe this could lead to family break up.
Lastly, Allaah has clarified how to treat a disobedient wife and this is explained in Fatwa: 83324. We hope that the above advice and guidelines are beneficial and lead to reconciliation.
Allaah knows best.