The conflict in "A Party Down at the Square" is internal. The protagonist, a white man visiting from Cincinnati, goes along with his Uncle Ed to "a party down at the square." Little does he know that he's about to witness his first (and last, according to him) lynching. Even though he uses the n-word freely enough, burning a man to death is new to him. He watches with fascination but he doesn't seem to understand why it's happening. He doesn't know what the victim has been accused of, even, and he is ultimately unable to handle the horror that unfolds as the man almost burns to ashes. He tries to leave but is hemmed in by the crowd, and when he finally breaks free, he throws up. He observes in the last paragraph that "Uncle Ed said they always have to kill niggers in pairs to keep the other niggers in place," but the narrator admits he doesn't think it's working, since they all come back and "look mean as hell when you pass them down at the store."
The moment of crisis occurs when a plane, its pilot confused by the brightness of the fire, starts circling overhead. The narrator realizes that the plane is about to crash, causing the assembled lynch mob to scatter. There is a period in which we think the lynch victim may get away while the mob is distracted or the narrator might (?) free him, but the victim is tied securely and beginning to roast over the growing fire. The mob then returns to their victim and throw gasoline on the fire, continuing the torture.
A peripety is a turning point in which the protagonist--in this case, the narrator--changes in some way we don't expect. In this story, we get a hint of this at the end, after the narrator seems to be having a revelation regarding the horror and unfairness of lynchings. In the same paragraph, as though these details were just as important and thought-provoking as the lynching he witnessed, he relates how a sharecropper spit tobacco on Brinkley's floor because the store keeper wouldn't give him credit. The incident is not as important, but we realize at this point that, despite the narrator's initial horror and revulsion, the lynching he witnessed is just another cool story, and he ultimately doesn't really care.