Gemara

(Rav Yehudah): The Mishnah obligates men and women to do every Mitzvah incumbent on a son to do for his father.

(Beraisa): "A man (will fear his mother and father)" obligates only a man. "Tira'u (you (plural) will fear)" obligates also a woman.

It says "a man" because a man can fulfill the Mitzvah. A (married) woman cannot.

(Rav Idi bar Avin): If she was divorced, she is obligated just like a man.

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

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

Answer (R. Yehoshua): Put the water in front of them. (Neither has precedence.)

Rishonim

The Rif and Rosh (1:44) bring our Gemara. (The Rif omits the question and answer about precedence of honoring the father or mother after divorce.)

Ran (DH Ish): One might have thought that once a woman married, her Mitzvah was (permanently) uprooted. Rav Idi teaches that this is not so.

Rambam (Hilchos Mamrim 6:6): Men and women are equally obligated to fear and honor parents. However, a man can fulfill this, but a woman cannot, because other authority is over her. Therefore, if she was divorced or widowed, they are equal.

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.

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).

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 for her husband even if she has many slaves. This is not clear.

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. The Mitzvah mid'Rabanan to fulfill the words of the Mes applies only when he gave something over. Even the dissenting opinion says that it is a Mitzvah only when the command was given directly to the one who must fulfill it. However, the daughter has a Mitzvah mid'Oraisa to honor her father! 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 is obligated to 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 has precedence over 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 supports the Beis Yosef. He said that R. Yehoshua told the orphan 'you do not ask to apply in practice, rather, 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.

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.

Shulchan Aruch (17): Men and women are equally obligated to fear and honor parents. However, a man can fulfill this, but a woman cannot, for she is obligated to her husband. Therefore, she is exempt from Kivud Av v'Em while married. If she was divorced or widowed, she is obligated.

Prishah (25): Even though even a man need not finance his parents from his own money, rather, (just to serve them) with their money, a woman cannot even serve them bodily, for she is always obligated to work for her husband.