He might be lying... but kotori telling her father not to tell him about her bruise makes it seems like he's not...
but last chapter really gave the dad an abusive kind of look
MAH BRAIN DON'T WORK BUBBA!
...and what do her bruises have to do with not reporting her as missing?
However, given that we already got word from that girl from school how Hatsushiro was apparently lying about her school to Yuuki, it makes more sense that her reason for running away/attempting suicide was bullying in school and not an abusive parent. Though it could still be both.
Yes. It is the father who abused her. I forgot but seems like her mother death is related to her and her father went insane and kinda blame her after that.