Squeaky is the protagonist and narrator of "Raymond's Run" by Toni Cade Bambara. We know a lot about her relationship with her brother, Raymond, but we know very little about her relationship with either of her parents. In fact, we know everything the author wants us to know about the relationship in two sentences found in the first paragraph of the story.
Hazel Elizabeth Deborah Parker, known as Squeaky, says this:
I don’t have much work to do around the house like some girls. My mother does that. And I don’t have to earn my pocket money by hustling; George runs errands for the big boys and sells Christmas cards. And anything else that’s got to get done, my father does. All I have to do in life is mind my brother Raymond, which is enough.
So, what we know is that Squeaky's mother does not make her do any chores like most of the other mothers do with their daughters; her mother does them herself. She obviously does the chores in exchange for one job--taking care of her mentally challenged brother, Raymond.
You ask about the relationship between mother and daughter, and that is difficult to ascertain just by these two facts. Perhaps Squeaky's mother is incapable of taking care of Raymond, or perhaps she knows Squeaky is better suited to the task. Perhaps there is something more selfish or unkind involved, but that does not seem likely.
Whatever the reason for this kind of role reversal, it seems to suit Squeaky just fine. She has figured out a way to care for Raymond and truly seems to care about him. And, even with the difficult and time-consuming task of caring for Raymond, Squeaky finds the time and energy to do her own work and accomplish her own goals--with a positive attitude. However odd it may seem to us, her relationship with her mother seems perfectly acceptable to Squeaky. More importantly, the fact that the author reveals so little about Squeaky's mother suggests that it must not matter very much to the story.