Yeah, Minami's dad was an abusive drunk who came after her with a knife.
Note that this would not be incompatible with the parental murder-suicide story Shizuka describes (though realistically she gave her own horrible backstory to establish shared ground.)
Shizuka found out, intending to lord her backstory over her, but realized she sympathized, so she is sharing her own backstory because she feels like Minami can relate.
That's a pretty debatable reading, as her self-described motive for prying rather appeared to be genuine curiosity about Minami...
So holy shit. Shizuka's mom pulled a murder suicide and Shizuka had to bury the bodies. That is so dark.
Goes ways to explain why she's so chronically fucked up, certainly.
On another note, was I the only one to notice that Shizuka's version (recounted internally ie. without any need to gloss things for the benefit of the target audience) of their initial acquaintance indicates the one Minami gave Iori way back in ch 12 is basically BS? Not that it's hard to see why she'd prefer to tell a take that's both rather compressed and "lighter and softer"...
It does, however, have the unfortunate side effect of rendering anything Minami tells about her past inherently suspect at least as far as details are concerned. :/