If mother and father fight with each other (whether physically or verbally), whom should child side with?
The Gemara in Kiddushin (31b), in explaining how to fulfill the Mitzvah of fearing one's parents, says that, among other things, he should neither contradict them nor support them. Rashi explains that the latter is called fearing them so as not to make it seem like your parents need your support. Based on that, I'd advise sitting this argument out.
If the child has to get involved, I'd try to side with both parents. Be a "talmid of Aharon, loving peace and chasing after peace" (Avos 1:12). We're told that "Hashem couldn't find a vessel to hold Brachah other than peace" (Uktzin 3:18). These are among the many sources that describe the power of peace. If he's forced to do something, he should try to compromise between them, unless it would make things worse. The best course of action would be for the child to go find the Rav of the community and ask him to sit down with the parents for a round of marriage counseling.
If parents fight the child has an obligation of kibbud av va'em to give the parents the impression that their fighting did not have a deleterious effect on the child.
The parents' fighting is likely to be due to a negative infringement from the 'outside world' which they try normally to protect their child from within the environment of the family.
Therefore if the child shows they are deleteriously affected by the fighting they will make the parents feel they have failed in this duty.