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According to the laws of respecting your parents, a child may not tell his parent that he/she is wrong. (Shulchan Aruch, Yoreh De'ah 240:2)
Now, if both father and mother are arguing about something and the child feels that he has some insight in the matter, may he weigh in and help his father win the argument?
What about vice versa?
A person cannot be מכריע his parent's words, even to say נראין דברי אבא, I agree with my father. See Yoreh De'ah 240 siff 2. As such, simply agreeing with one's father is not allowed. This has nothing to do with who the father is arguing with, the fact that it is the child's own mother that the father is arguing with doesn't make it better.
The gemoroa states that is someone is faced with a situation in which his father is wrong, he should say "didn't you teach me ..." or in some other fashion bring it up without explicitly contradicting him.
In this case also, perhaps he should ask "Is such and such a source relevant to what you are discussing?"
However, this is a lose lose situation and the best thing might be to stay out of it.
A child is obligated in כבוד, honor (Shemot 20:12) and מורא, fear (Vayikra 19:3) of his parents.
This is brought down by the שלחן ערוך, YD 240, who defines מורא (se'if 2, there) as not doing disrespectful things toward his parents -- not to stand in their places, not to disagree with them, etc. כבוד is defined (se'if 4) as positively doint things to honor one's parent -- feeding them, clothing them, etc.
שלחן ערוך rules (14) that:
אָבִיו אוֹמֵר לוֹ: הַשְׁקֵנִי מַיִם, וְאִמּוֹ אוֹמֶרֶת: הַשְׁקֵנִי מַיִם, מַנִּיחַ אִמּוֹ וְעוֹסֵק בִּכְבוֹד אָבִיו. [...]
If one's father says to bring him water, and one's mother says (at the same time) to bring her water, one must give water to one's father. [...]
Note, though, that per the definitions above, this is not a case of מורא, but rather a case of כבוד. As far as I could tell, there is no discussion about a conflict between the מורא of one's parents in the שלחן ערוך, but I would suggest that it shouldn't be too different fromt the case of כבוד, and one should side with one's father.
In a more practical way, I would echo what sabbahillel said: "However, this is a lose lose situation and the best thing might be to stay out of it." -- in a real-world application of this הלכה, one should try as hard as possible to avoid picking sides, using this only in a situation where there are no other options.
although the abovementioned gemara in kiddushin says if your mother and father both ask you to bring them a cup of water, you should listen to your father, I would assume that your father only takes precedence where you can only fulfill one of the mitzvos kibud at this time.
in this case, you can fulfill both mitzvos of מורא at the same time: by staying out of it and not contradicting your father or mother.
as a side point, respectfully pointing out a new insight into the discussion is not the same as telling your parents that they are wrong.
