The only reference to Mitsuki's family I can find is a photo from the first page of Rhythm B, where she's posing with a man in glasses who might be a dad (or maybe Joe just used to wear glasses). As she's leaving somebody yells at her to "leave the headphones here!" but there's no hint at who said it. She's leaving a house with the family name outside.
Maybe Im reading this part wrong "Joe just used to wear glasses" since english is not my native language but Joe, since the start of the manga, was using glasses; it's now from an illustration first and this chapter that we can see him without it.
And the Kanji that we see in the first page of Rhythm B says "Elemetary School" (I think this is what is called in english, if not, the last kanji is the one for "school").
Just saying this to say that indeed we don't really have strong evidence of what are Koga´s parents situation, just Koga in chapter 8 when arrives to her home, we can see from her attitude that "no one cares that she is in home"; but it doesn't gives us anymore info from what we know now