Gemara

(Mishnah) Suggestion: The Torah (almost) always mentions the father before the mother. Perhaps the Mitzvah to honor a father is greater than the Mitzvah to honor a mother!

Rejection: "Ish Imo v'Aviv Tira'u" teaches that they are equal. However, Chachamim taught that the Mitzvah to honor a father takes precedence, because also the mother is obligated to honor him.

Kidushin 30b (Beraisa - Rebbi): Hash-m knows that a child honors his mother more than his father because she entices him with things. Therefore, He commanded to honor the father before the mother. Hash-m knows that a child fears his father more than his mother, because his father teaches him Torah. Therefore, He commanded to honor the mother before the father.

Question (an orphan): If both my father and mother ask me to bring them water, which request should I honor first?

Answer (R. Eliezer and R. Yehoshua): Your father's comes first, for you and your mother are commanded to honor your father.

Question (the orphan): If she is divorced, what is the law?

Answer (R. Yehoshua): I see that you are an orphan. Put the water in front of them.

Rishonim

Rambam (Hilchos Mamrim 6:2): When the Torah commanded to honor parents, it mentions the father before the mother, and in the command to fear parents, it mentions the mother before the father. This teaches that they are equal for honor and fear.

Kesef Mishneh: The Rambam learns from Rebbi.

Bach (YD 240:2): The Rambam and Tur did not explain how we learn that both are the same for both. Perhaps for honor the father has precedence, and for fear the mother has precedence, like the verses connote! Semag (Aseh 112) explains that the Torah mentioned the father first regarding honor, for one leans to honor his mother more. The Torah mentions the mother first regarding fear, for one leans to fear his father more. It seems that the Rambam and Tur agree, just they did not elaborate.

Rambam (14): If one's father and mother both asked him to bring water, he desists from honoring his mother and (first) honors his father, because he and his mother are commanded to honor his father.

Poskim

Shulchan Aruch (OC 240:14): If one's father and mother both asked him to bring water, he desists from honoring his mother and (first) honors his father.

Ha'Makneh (Kidushin Sof 30b): The father has precedence for honor (e.g. serving food and drink, transporting...) Regarding fear (not sitting in their places, not contradicting them...) there is no precedence, for a woman need not do these for her husband. She only needs to serve him through the chores that Chazal fixed.

Pischei Teshuvah (9): Perhaps if his father and mother both requested a chore that a woman need not do for her husband, they are equal. The Gemara discussed bringing water, for a woman must do this for her husband even if she has many slaves. This is not clear.

Shulchan Aruch (ibid.): If his mother was divorced from his father, he honors first whichever he wants.

Beis Yosef (DH u'Mah she'Chosav v'Im): Since we explained why the father has precedence, the Rif and Rambam did not need to teach that after divorce they are equal, and he may honor first whichever he wants. R. Yerucham says that he must put the water in front of them and tell them to come and drink. He may not give precedence to one over the other. It seems that Rashi (31a DH she'Ben, 'the honor of both is equal upon you') holds like I wrote.

Pischei Teshuvah (12): The Yam Shel Shlomo (Kidushin 1:62) rules like R. Yerucham. However, his mother has precedence to be redeemed from captivity, and if they request food and clothing, for normally women have precedence over men for these (YD 251:8).

Yam Shel Shlomo (Kidushin 1:63, cited in Pischei Teshuvah 14): Some say that if one's father told him not to say Kaddish for his mother, he obeys, for the father's honor comes first. He should not obey (like the Rema below). There was an enactment to say 'I am an atonement for my parent' in the year of Aveilus. All the more so it is a Mitzvah mid'Rabanan to say Kaddish, which helps the Mes tremendously. Not saying it transgresses "v'Ahavta l'Re'echa Kamocha". However, one obeys a father's command not to wear black when mourning over his mother, which is a mere custom.

Noda bi'Yehudah (2 EH 45): A case occurred in which Levi, before he died, commanded his wife not to allow his adult daughter to marry his brother. A Rav said that death is unlike divorce, for a widow must honor her deceased husband. This is wrong. She may remarry! However, if one's father commanded him to do something and died, and his mother protests, he heeds her, for honoring in life has precedence over honoring after death.

R. Akiva Eiger (1:68): If one's mother commanded him to do something and his father protested, he may fulfill her request if refraining would cause a monetary loss, for we hold that one need not use his money for Kivud Av. Even if there will be no monetary loss, this is like divorced parents who both requested water. He may fulfill either first, for she need not honor his father. The same applies after his mother died. Death separates like divorce. Livyas Chen similarly says that one must honor his grandfather more than his father only while both are alive, for then also his father must honor his grandfather. However, the Beis Yosef (YD 376) says that if a woman commanded her son to say Kaddish for her after her death, and his father protests, Kivud Av has precedence over Kivud Em. The Noda bi'Yehudah says that honoring in life overrides honoring after death, even if the father died and the mother is alive. This is unlike the Beis Yosef, who gives precedence only because Kivud Av is greater. Rashi (31a DH she'Ben) supports the Beis Yosef. He said that R. Yehoshua said 'I see that you are an orphan. I.e. you do not ask to apply in practice. You just seek to learn.' If death is like divorce, perhaps he needed to know about a request after a parent died, and the living parent protested! Rather, it is clear that even after death, the father has precedence.

Pischei Teshuvah (YD 240:14): If a father tells his son 'do not say Kaddish for your mother', he should not obey, for the father transgresses "v'Ohavta l'Re'acha Kamocha."

Birkei Yosef (OC 38:7): The Tur connotes that that the father and mother are both in front of him at the same time. If one began honoring one of them, he is exempt from honoring the honor (until he finishes honoring the first), even though others cannot do (honor) the latter.

Question: In a certain congregation, if in a week one member has Yartzite for his father, and another has Yartzite for his mother, the former has precedence to read the Haftorah on the previous Shabbos. If a Chacham has Yartzite for his father, and a Ba'al ha'Bayis has Yartzite for both parents that week, who has precedence?

Answer (a Chacham cited in Birkei Yosef OC 286:1): There is no source for the custom. The Levush (OC 130) brings from Maharam Mintz (80) that father and mother are equal. If parents are divorced, there is no precedence to honor one over the other. Rashi says that he honors first whichever he wants. The Rif and Rambam hold that he may not honor one over the other. He must honor them at the same time. The Shulchan Aruch rules like Rashi. I say that the two who have Yartzite should compromise. However, here the Ba'al ha'Bayis has precedence, for he has two Yartzites. A Chacham has precedence in worldly matters, but not in spiritual matters, like Ra'anach (2:70) says. Perhaps a Ba'al ha'Bayis always has precedence, for the child's Mazel is bad at the time of the Yartzite, and a Chacham's Torah protects him. A Chacham brings great merit to his parents through his Torah. The Ba'al ha'Bayis does so only through Kaddish and the Haftorah. He is like an Oni (poor person), who has precedence to receive Tzedakah!

Birkei Yosef: If the Rif and Rambam argue with Rashi, why would the Mechaber rule like Rashi? Whenever two of the Rif, Rambam and Rosh agree, he rules like them against (even many) Poskim! Further, the Beis Yosef explicitly said that the Rif and Rambam hold that they are equal, and one may choose whom he wants to honor first. R. Yerucham holds that one may not give precedence to either, but he is alone against the Rif, Rambam and Rashi. Perhaps the custom is to give precedence to a father's Yartzite is because nowadays, most men sin through wasting seed and Bitul Torah. They need merits of their children more than women do. Alternatively, men are commanded in more Mitzvos. This is a reason for Beis Din to hear first a case of men (Yevamos 100a, although there embarrassment of women overrides this). The Rambam and Bartenura explain that this is why men have precedence for sustenance and returning Aveidos (Horayos 13a). We rule like this Stam Mishnah. When there is no issue of shame, we honor men first. Regarding Kivud Av v'Em, neither parent has precedence if they are divorced. Here is different, for two different people want to honor their parent. A Chacham has no precedence over a Ba'al ha'Bayis. It depends only on the deceased.

R. Akiva Eiger (ibid.): When he need not heed his father's protest for it will cause a loss, he need not fulfill his mother's request. There is a Mitzvah to heed his father, just he is not obligated to do so. If he does, it is as if he gives a gift to his father (of the money he loses) and fulfills Kivud Av properly.

Rema (376:4): The custom is to say Kaddish for one's mother, even if his father is alive. The father cannot protest.

Magen Avraham (OC 132:2): The Rivash (115) says that if a man is adamant that his son not say Kaddish for his mother, the son must obey, for honoring the father has precedence. The Rema rules unlike this.

Teshuvah me'Ahavah (in Yalkut Meforshim in Shulchan Aruch ha'Shalem): The Birkei Yosef says that after she died, it is as if she was divorced, and Kivud Av and Kivud Em are equal.