The primary conflict is how parents effectively justify their children fighting and dying in the nature of war.  The main characters are a husband and wife who are in a railway car.  The husband is explaining to the other passengers in the car that his wife is distraught at their son fighting in the war.  The other primary character is a fat man who enters the train and offers his own analysis which is that parents have a responsibility to encourage their kids to fight for the nation and embrace the consequences of dying for national glory.  He pontificates quite a great deal and is only disarmed by the wife's emotionally and rhetorically jarring question at the end of the story.  The fundamental conflict of being able to stand by a nation while there is obvious conflict on an personal level of what is being done in the name of said nation becomes the overarching dilemma in the story.

To me, the main conflict in this story is between the various people on the train.  They are arguing about how they should think of the war.  They are also arguing about who should be saddest and whose situation (in terms of sons going off to war) is the worst.

I think this symbolizes a deeper conflict, though.  It is a conflict between the people and society.  The characters (they are all just anonymous people who have sons who have been called to fight or who have already been in the war) are showing the conflict between individuals' desires to have their own happiness and society's need to have its citizens fight for it in wars.

In the short story "War," who is the protagonist and who is the antagonist?

Oddly, Nobel Prize winning (1934) dramatist Luigi Parandello's short story "War" has no specifically designated protagonist or antagonist; instead all the characters who board the train at Fabriano are both protagonist and antagonist to themselves. Whether they argue for the futility of war and the senseless waste of life or the heroism of serving one's country in their human need for value, they seek some alleviation from their desperation, but no one gives this respite to them. 

With such a minimized setting, Parandello intensifies the focus upon the unimpressive characters, among whom some possess eyes that are "watery and motionless." When the heavy man who hides his missing teeth with his hand contradicts the sentiments of the others by declaring that he wears no mourning clothes for his dead son because he honorably served Italy in the war, dying satisfied with his life--“Our children do not belong to us, they belong to the Country”--,the heavy woman who has sat silently all the time, lost in the single fear of her only son's having been called to serve, raises her head and listens.

It seemed to her that she had stumbled into a world she had never dreamt of, a world so far unknown to her and she was so pleased to hear everyone congratulating that brave father who could so stoically speak of his child's death.

In a climactic moment, shaking herself from her brown study, she raises her head and asks, "Then...is your son really dead?" Ironically, then, the man's role changes from that of an antagonist to the others' sentimentality to that of a protagonist faced with the adversary of reality stripped of its heroics and illusions. Suddenly, this man of "watery, light gray eye" breaks into heart-wrenching, uncontrollable, distressing sobs.