Praise be to Allah, and peace and blessings be upon the Messenger of Allah and his family.
The situation in which you are, dear brother, is certainly a difficult one. It is not easy to live with a father that has aberrations and perversities stemming from being misguided from the true path of ahl us-sunnah wal-jamaaah (the followers of the Quran, sunnah, and our pious predecessors). However, a Muslim is to anticipate and seek reward from Allah for: being patient with and persevering in dealing with your father; and, being polite when counseling and showing him the truth of guidance, using appropriate and agreeable methods that do not give your father the impression that you are trying to show superiority or arrogance, or that you are belittling him. Instead, your father should feel that the advice is from a son that acknowledges and respects his position as his father, and that your concern is out of care and veneration., as the case was with Prophet Ibrahim (peace be upon him) when calling his father to the path of truth. Allah the Most Glorious said in the Quran (interpretation of the meaning):
When he (Ibrahim) said to his father: "O my father! Why do you worship that which hears not, sees not and cannot avail you in anything? O my father! Verily, there has come to me of knowledge that which has not come to you. So follow me; I will guide you to a straight path. O my father! Worship not Satan. Verily, Satan has been a rebel against the Most Beneficent (Allah). O my father! Verily, I fear lest a torment from the Most Beneficent (Allah) overtake you, so that you become a companion of Satan ( in Hell-Fire)." He (the father) said: " Do you reject my gods, O Ibrahim? If you do not stop (this), I will indeed stone you. So get away from me safely before I punish you." Ibrahim said :" Peace be upon you. I will ask forgiveness of my Lord for you. Verily, He is unto me, ever most gracious. (surat Maryam, 19:41-47)
Note that Prophet Ibrahim (peace be upon him) addressed his father in the most subordinate and most refined, courteous, and polite expression (abati in Arabic). Prophet Ibrahim (peace be upon him) did not say to him, "I am a scholar and you are ignorant"; instead, he said "There has come to me of knowledge (as a Prophet) that which has not come unto you." Ibrahim (peace be upon him) showed his concern and compassion towards his fathers well-being when he said, "O my father! Verily, I fear lest a torment from the Most Beneficent (Allah) overtake you." When his father rejected the truth and threatened to stone him, Ibrahim (peace be upon him) replied in utmost politeness, "Peace be upon you," and promised him that he will pray to his Lord on his behalf for forgiveness. This is how a righteous son should advise his misguided father.
You also should know that rejecting or denying the sunnah of our Prophet (peace be upon him) is an extremely grave matter; hopefully this subject will be covered in detail as a separate issue. In brief, if your fathers bidaah (innovation in religion) takes him out of the fold of Islam, then your prayers behind him is not valid since he would be a kafir (infidel, unbeliever). Such would be the case if, for example, he insists and persists in completely rejecting the sunnah in its entirety, even after the proof and evidence demonstrating and substantiating the truth and validity of the matter is presented to him. On the other hand, if your fathers bidaah does not take him out of the fold of Islam, such as the case of refusing to perform certain acts mentioned in the Quran and Sunnah, then it is permissible to pray behind him and your prayers are sound, and Allah knows best.
Addendum: Sheikh Muhammad ibn Salih ibn Uthaimeen has replied the following with regards to this question:
Denying (the sunnah) could be of two types: a denial with regards to interpretation or a denial of rejection. If it is a denial of rejection, in the sense that one might say, "Yes, I realize that the Prophet (peace be upon him) said this, but I reject it and dont accept it," then the person is a kaafir (unbeliever) and has committed apostasy. Thus, it would be impermissible to pray behind him. If, on the other hand, it is a denial of interpretation, then he is granted a respite, if the interpretation is possible and warranted by the language, and he knows the sources of Islamic law and its resources. In this case, it is not an act of kufr, but rather under the category of those who innovate incorrect practices in the religion (if his interpretation is such). It would be permissible to pray behind such a person, unless there is a beneficial objective behind not doing so, such as deterring him from such opinions, so that he may reconsider his mistaken interpretations. In this latter case, then it would be preferable not to pray behind him (if such a beneficial effect would be realized).
