I think "elle a dû s'occuper de sa sœur" would also be translated as "she had to take care of her sister", but you couldn't use it in that sentence. The passé composé implies that she actually did take care of her sister, and that is now completed. That's not consistent with the aspect in the beginning of the sentence: "Now that her parents were gone" which sets the scene just after her parents have gone, and while she is still taking care of her sister.
I think the difference is that the sentence as written in French in imparfait is what would be used if it was unknown whether she looked after her sister or known that she did not, and therefore in French informs us that it is an unknown. If it was written in passé composé, we would know that she had (very likely anyway) looked after her sister. I am not sure that it is correct to say that «maintenant» changes that - it helps define the time at which the obligation started. The obligation clearly arose in the past as the sentence only uses past tense verbs. It depends on what is known by the speaker. An important point here is that the different past tenses in French relate different information, even if when translated to English that implicit French meaning is lost.
When describing an event in the past, you can often choose either the passé composé or imparfait depending on how you want to describe it. Of course the word "maintenant" does not mean that the events are happening at the time they are being narrated, but it makes the reader/listener imagine they are present at that point in time.
I think the instruction of this lesson perhaps gives a clearer meaning to this than the translated example. ie - use of "supposed to"
....she was supposed to take care of her sister.
This introduces much more doubt about whether she actually did or not. In fact, if I were reading this in a story I might expect the next few sentences to explain how she didn't look after her sister.
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