Again,... didn't Sangwoo once say to Bum that, before that one old man they killed together, he hadn't killed a man since his dad (that before the old man, he had only killed women ever since)? That means he was the one who did it, or at least he participated. Taking into account that his mom also wanted a secret for herself that she didn't have to share with either Sangwoo or with his dad (in the same way as they both were keeping stuff from her) she can't be talking about killing Sangwoo's dad, since Sangwoo would have to have known about it too, if he did indeed help her with it (and if, as we have established, for that earlier conversation with Bum to make sense, he had to have at least participated).
I think her secret is that she killed the father....just saying djskskej