IMHO C
Here is why
(A) This is illogical ; ask the question who was looking through the expensive binoculars. Definitely the turkey wasn't looking through the binoculars, perhaps some person. Since the object being modified by the participle phrase (when in the beginning of the sentence) must follow the participle phrase ( which is not the case in A), we can eliminate A.
(B) Same issue as in A. Ask the question : who was using the expensive binoculars? Therefore, eliminate B too.
(C) Through the expensive binoculars is an adverbial modifier ( It is a prepositional phrase ( through is a preposition) that is acting as an adverb). Remember : Adverbs have much looser placement rules as compared to adjectives such as participle phrases.
Be careful: prepositional phrases can be adverbs as well as adjectives. But what separates them apart is that adverbs answer questions such as how? when? where? .
We can confirm that through the expensive binoculars is an adverb because it identifies from where the turkey was seen.
Since this phrase - through the expensive binoculars - is an adverb,it must be modifying an adjective, an adverb, or a verb (by definition). The verb -appeared- is correctly being modified by the adverb phrase : through the expensive binoculars.
Also, you can further confirm this by asking the question - from where did the turkey appear to be the size of a robin? You will find that "through the expensive binoculars " is correctly answering this question.
(D) While can be used in 2 situations: first, for indicating contrast (when used in a subordinate clause0; second, for indicating that 2 actions happened at the same time ( when used as an adjectival modifier).
In this situation (the actual sentence) we can be assured that no contrast is going on.
This sentence however conveys that a contrast is happening. It seems to say " At the same time the turkey was being watched by ... through ... the turkey somehow (tried) to appear to be the size of a robin". In other words it seems to say that 2 events happened.
For eg. While I watched TV, my friend played on the PC. ( two events : me watching TV and my friend played on PC).
This is wrong because only 1 even happened in original sentence) and that was of seeing turkey, which appeared to be the size of a robin.
The original sentence wanted to convey that the turkey , seen through binoculars, appeared to be of the same size as that of robin.
(E) It is an participle phrase. It is a dangling modifier because the turkey vulture cannot look through binoculars and appear to be the size of the robin (not logical).
Remember E is slightly different from D in that D had a subject (the hiker) and a verb (was being watched) and therefore acts as a subordinate clause. The preposition while in the subordinate clause was used to indicate contrast as in usage of although.....,....Therefore, " while... binoculars" isn't really modifying the turkey ....robin.
But in E there is no verb. looking is a gerund here and through the expensive binoculars is a prepositional phrase.
More specifically E is an example of two things happening at the same time.
for eg: While watching TV, I ate my lunch. ( watching and eating are happening at the same time).
