Though this sentence is long and not necessarily in your favorite style, there is nothing grammatically wrong with it.
Stripped way down, the last part of this sentence would read: "The interiors were overcrowded, their floors serving as beds and their walls windowless." The last two phrases modify the interiors, giving more information about the insides of these apartments. Joined by "and," they must be parallel to each other.
It may seem odd, but here the word "serving" in the phrase "their floors often serving as beds" actually functions as an adjective. This is fairly common, however. For example, in the sentence "She ran for the bus, shouting, and finally caught it," the verbs "ran" and "caught" are parallel, while the word "shouting" functions as an adjective, describing what she was like while "she ran for the bus." Thus, each of these modifying phrases is noun + adjective, and they are parallel in construction.
Since there is nothing wrong with this sentence, just check each of the other choices to see if any of them is better stylistically.
Scan and Group the Answer Choices:
There is a clear and useful 3-2 split here. (A), (B) and (C) all begin "whose interiors were" and (D) and (E) both begin with "having interiors."
Eliminate Wrong Answer Choices:
(D) and (E) both begin with the word "having," which is incorrect, because it isn't clear who or what has these interiors. Is it the apartments, the immigrants, or Jacob Riis? Such ambiguity is unacceptable, which is enough reason to rule out these two choices. Further, (D) changes "dilapidated with age" to "dilapidated due to age," and (E) changes it to "dilapidated on account of age." Both are wordier than the original. In fact, the phrase "on account of" is usually wrong on the GMAT. So, both (D) and (E) are incorrect.
(B) makes several small changes from the original. First, it changes "floors often serving" to "floors were often serving." This small change has big implications. Without the word "were," the phrase "their floors often serving as beds" is a descriptive aside. With "were," this is an independent clause (it could stand on its own), so it would at the very least have to be parallel with the preceding clause, making a list with it. However, it isn't parallel; to be parallel, it would have to read, "whose floors served as beds." (B) also inserts the word "were" in the last clause, which creates exactly the same problem there. Finally, it unnecessarily changes the phrase "dilapidated with" to "dilapidated due to." "With" is fine, so there's no reason to make the phrase any longer. Because of these problems, (B) is incorrect.
(C) also creates an unparallel list. To be parallel, the second and third clause would have to begin "whose walls" and "whose floors."
Since each of the other choices introduces new errors, the sentence is correct as written and Answer Choice (A) is correct.
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