cantaffordname
This is example of parallelism. also, the sentence is in past tense, so parallelism list needs to have past tense of verb. (in some case, parallelism can go have mix match of tense but those are rare cases and meaning is more important)
Errors are highlighted in Bold
The Parthenon was a church from 1204 until 1456, when Athens was taken by General Mohammed the Conqueror, the Turkish sultan, who established a mosque in the building and used the Acropolis as a fortress.
(A) who established a mosque in the building and used the Acropolis as - Correct, since established and used are parallel.
(B) who, establishing a mosque in the building, used the Acropolis like
(C) who, when he had established a mosque in the building, used the Acropolis like - No sequence, hence use of past perfect tense is wrong.
(D) who had established a mosque in the building, using the Acropolis to be
(E) establishing a mosque in the building and using the Acropolis as - No need of present perfect tense
Hello
cantaffordname,
A very good analysis of the problem, I must say. Keep up the excellent work.

I would like to add to your analysis the error about the usage of
like in
Choices B & C. The word
like is used to present either the comparison or the examples. But this sentence intends to convey that the Acropolis itself was used as a fortress. It means that the Acropolis = fortress. Since
the Acropolis functioned or acted as
a fortress, the sentence must use
as to present this intended function as we see in the correct answer choice.
Usage of
to be is incorrect in
Choice D for the same reason.
Thanks.

Shraddha