A) Barack Obama's breakthrough election of 2008 - Incorrect. Impact of African American vote in other elections is compared with Barack Obama's breakthrough election.

B) did Barack Obama's breakthrough election of 2008 - Incorrect. Impact of African American vote in other elections is compared with impact of Barack Obama's breakthrough election.

C) Barack Obama's breakthrough election of 2008 did - Incorrect. Same error as in B.

D) had Barack Obama's breakthrough election of 2008 - Incorrect. Tense error.

E) in Barack Obama's breakthrough election of 2008 - Correct. Perfectly compares the impact of African American vote in other elections with impact of African American vote in Barack Obama's breakthrough election.

Answer: E

While you mentioned that option D) has a tense error in this case, I would like to ask which usage , "did" or "had", in the below slightly modified statement is appropriate?

In no other election , except Obama's, did the African-American vote have such a profound impact as did/had the British-American vote in all other elections.

What am I missing here?

Well, D is not only a tense error, I'd say it is a comparison error (more important) + tense error.
Like A, D is incorrectly comparing 'Impact of African American vote in other elections' with 'impact of Barack Obama's breakthrough election', but additionally D uses two different tenses: have & had.

OFFICIAL EXPLANATION

This sentence construction problem requires a very careful look at the structures involved. The simplified sentence is this: "In no other election did X accomplish Y as in this election." The components that need to be parallel are "in no other election" and "in Obama's breakthrough election." Choices A, B, C, and D all lack the necessary preposition "in" required to make these structures parallel. Only answer choice E properly provides parallelism, so choice E is correct.
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