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I don't think it's Guwon who killed his father because If I remember correctly..Jinwoo's father was already dead before guwon came to rescue him. Jinwoo says he doesn't remember describing his father to Guwon, but Guwon saw a picture of them together when he first entered the house. Guwon is not a liar, we can see it, if he had killed him, he would have admitted it. + in the first chapter, city ppl were ready to Kill jinwon but Guwon saved him. + there's a part where we can see Jinwoo hurting someone. Also in the chapter 2, Jinwoo tried to kill Guwon (I am not sure if it's relevant to my opinion but—)
So there's two possibilities :
-the city people killed jinwoo's father
-Jinwoo killed his father.