A brief word from Ami
I am an only child, but then other than Usagi-chan most of us are only children. My parents divorced when I was young. My mother is a doctor and is often very busy at the hospital, so we don't see each other much. My father is an artist, and lives in France.
I am usually home by myself, when I'm home. If my mother is home, I will often go out to the library or to be with friends so that she can rest.
I do care a great deal for both my parents, but I don't see my father much, so we have unfortunately lost touch with each other in many ways. In other ways I sometimes worry that my mother and I have lost touch also. We communicate through notes, and through text messages, and not much in person at all. Sometimes I worry that she doesn't see me, but I'm afraid that I will disappoint her, or hurt her feelings if I say something about this. She works very hard, and it's very important what she does.
Documentation from the series
The mysterious Dr. Mizuno does have a couple of panels in SuperS in the manga where she apologises for not having been home recently and compliments Ami for being so good. From the brief appearance though we can see that Ami looks a lot like her mother, both having short dark hair, whereas her father in the one panel he has in the same shorts, is apparently blonde.
Dr. Mizuno makes little to no appearances in the anime, although she is given a brief flashback sequence in the first few scenes of R movie Ami is shown making cookies with her mother, and explains to the other senshi a way of imprinting the cookies that Dr. Mizuno taught her. So, perhaps her mother only went back to working quite a bit when she decided Ami was responsible enough to take care of herself. All of this is heresay, given it's based on one short scene and there is nothing really mentioned in the anime to back it up.
In the manga there is also evidence that Ami's family is quite wealthy, because after the Inner senshi visit the Moon Kingdom and bring back a sword, Minako chips a diamond of Ami's mother's but Ami tells her that, "It's okay, because her mother has lots of them."
In PGSM, Ami's mother is given a name: Saeko. She works at a teaching hospital, tied to the university. She focuses very much on Ami's schooling and that Ami do as well as possible so that she can be a doctor, during the course of the one episode she is in. It seems towards the end that she realizes Ami might only be studying to be a doctor because her mother wishes it, not because Ami wishes it, and she backs off, telling Ami she should do what she wants to do, rather than what Saeko wants.
In all incarnations of the show Ami's father seems rather irresponsible and uncaring, given he only communicates with Ami via picture postcards he draws and never writes or visits. Ami doesn't seem too view him too badly, despite this. Although she doesn't ever talk about him with the other senshi, that we see. He is given the briefest mention in PGSM, it's shown that he was still around when she was four because her parents apparently took her to a theme park together and let her ride the Merry-go-round.