What is the ruling on a Quran teacher going to teach children in their home in the absence of the father, with only the mother present, or in the absence of both parents, with the child and his sisters present in the home?

Private tutoring in homes is permissible according to Islamic law if it adheres to the Sharia-compliant guidelines. If the female tutor is safe from temptation and knows that the homeowner does not object to her entry, then there is no objection to teaching the student in a room separate from the women of the household.