As always, if we see that we have a modifier at the beginning of our sentence, we should look at the noun immediately after the modifier - a modifier always describes the noun closest to it, and putting the wrong word next to a modifier is a very common GMAT error.

Our modifier in answer choices A-C is "Recorded in New York city". The song "The Gift" is what is being recorded in this sentence, so it should be next to the modifiers. We can rule out A and B and hang on to C.

Answer choice D has two issues:
1) "recorded in New York city" is separated from its subject "The Gift" by a lot of different ideas, which makes the meaning of the sentence pretty confusing.
2) It isn't entirely clear which singer would later make her reputation as an actress - the modifier "who would ... actress" is next to Darlene, but because both sisters are listed before the modifier, it would make more sense for the modifier to refer to both sisters. C clears this up by listing the sisters in opposite order, so that the modifier can only refer to Darlene.
Eliminate D.

Answer choice E has the same issue with "who would ... actress", but it also has other issues:
1) The original sentence has "recorded in New York city" as the modifier and "written and sung by teenagers" as its main point. E switches these two, so that "written and sung by teenagers" is the modifier and "recorded in New York city" is the main point. This changes the focus of the sentence, which is a way to change the meaning of the sentence. The important thing isn't where The Gift was recorded - it's who recorded it.
2) The [singular] songwriter and [singular] singer is two teenaged singers - did only one of them write the song and only one of them record it? If so, which one did which?
3) Any answer with "being" should automatically be suspect - "being" usually indicates a lack of concision.
Eliminate E.

This leaves us with the correct answer, C.