+1 for C.

(A) by testing them quickly for a variety of illnesses and, before they expire, distributing them

(B) by testing it quickly for a variety of illness and distributing it before expiration

(C) by quickly testing it for a variety of illnesses and distributing it before it expires --> Correct, rectifies pronoun ambiguity and parallelism error

(D) by quickly testing it for a variety of illnesses and by, before they expire, distributing them

(E) by quickly testing it for a variety of illnesses and to distribute it before it expires

Hence, C.
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But expiration of what exactly : blood bank/ hospital/ donated blood ?
I just found it redundant.
Please correct me if I am wrong.
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KAPLAN OFFICIAL EXPLANATION:

First of all, the subject for the underlined portion is a donation. Since the subject is singular, the pronouns that refer to it need to be singular as well. Eliminate (A) and (D) for using they and/or them rather than it. Compare (В), (C), and (E). Eliminate (E) because testing and to distribute are not parallel. Since both are in parallel situations, each describing what the blood banks need to do, they need to have the same structure. Finally, (C) is better than (B) because quickly testing it is more direct than testing it quickly and because before expiration at the end of (B) is less clear than before it expires. Go with (C).
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Expiration = Noun
Expire = Verb

Here we require a Verb thus "Expire" is needed, hope this helps !!!
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