Whatever noun that comes after the first dependent clause has to be the noun that was 'looking through the expensive binoculars. In the original sentence, the turkey is looking through them.. which does not make sense. Using this logic, we can eliminate A, B, and E.

Which leaves C and D. D sounds really wordy and has an unneeded noun "it" to describe the turkey. The "the" also doesnt seem to be needed... On top of that, it introducs a new noun- the hiker.

C is much more concise and retains more of the meaning than option D.

Answer is C.

(C) Through the expensive binoculars,

Looking through the expensive binoculars, the turkey vulture appeared to be the size of a robin.

(A) Looking through the expensive binoculars, (underlined portion modifies the turkey vulture, so wrong)
(B) Using the expensive binoculars,(same problem as in A)
(C) Through the expensive binoculars,(Correct)
(D) While it was being watched by the hiker through the expensive binoculars,(Wordy and awkward)
(E) While looking through the expensive binoculars,(same problem as in A)

IMO: C

since sc test meanings, new words is problematic for non natives like me. I have to consule dictionary to know the words "vulture" and " binocular" before doing this one.

C is wrong because it add the meaning of time presented by "while" . we do not say in the time we see it through the binoculars. we need just to say that using the binacular, the vulture to be the size

hard one.