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.