A. The commandment of honoring parents is one of the Ten Commandments (Exodus 20:12):
Honor your father and mother. You will then live long on the land that God your Lord is giving you.
The Torah equally commands respect, or awe, of parents (Leviticus 19:3):
Every person must respect his mother and father, and keep My Sabbaths. I am God your Lord.
While the order of father and mother is different in these commandments, neither distinguishes between the two. The same is true for other commandments relating to respect for parents, such as the special stringency of striking or cursing parents (Exodus 21:15,17).
Rashi's commentary on the verse in Leviticus explains that the difference in ordering is in fact meant to accentuate the equality of the parents: even though the natural tendency is to honor the mother more and respect the father more, the Scripture presents it in the opposite order to emphasize that the commandment applies equally to both.
The only time Jewish law gives preference to one parent is when there is a "two birds with one stone" situation. The Talmud states:
The son of a widow asked Rabbi Eliezer: If the father says "Get me some water" and the mother says "Get me some water", which one has precedence? He replied, defer the honor of your mother and do the honor of your father, for both you and your mother are obligated in the honor of your father.(1)
But this is due to a particular confluence of events: since in the usual household roles (at least during the time of the Talmud) it was the wife's responsibility to take care of feeding the husband, in this particular instance giving water to the father honored both him (by fulfilling his request) and the mother (by taking care of her responsibility on her behalf). So honoring the father comes first. The passage then goes on to explain that if the parents are divorced this preference disappears, and neither parent has precedence. In this case the child can choose which parent to take care of first. The prominent early commentator Rabbi Menachem HaMeiri states that in that case the child should give precedence to whichever help he can give most promptly, so as to show alacrity in carrying out God's commandments.
In fact, we could imagine a situation where the same rule applied in the opposite manner. Again, in a household with traditional gender rules it is the man's job to take care of repairs. If the mother said "Please fix the faucet" and the father said "Please fix the boiler", the mother's request would logically have precedence since both the son and the husband are obligated to take care of the mother's repairs.
If the difference of opinion applies to your personal conduct and not to the duties you owe your parents, you are not obligated to listen to either parent. Carefully considering a parent's advice is respectful and in most cases wise and prudent, but it is not what the Torah is primarily referring to. Honoring and respecting parents is first of all about their needs, not yours; you and not they are responsible for choosing your conduct in everyday life.
Differences of opinion among parents can be merely inconvenient or may be traumatic, and I hope that you will be successful in maintaining constructive and loving relations with both parents. To the extent that you have to choose, you should know that the Torah does not give precedence to either parent in the commandments of honor and respect.
SOURCES: (1) Babylonian Talmud Kiddushin 31a