IMO E is the correct ans
the s/c is testing comparison, and rule is the items being compared must be alike
original i.e A is comparing the public's apetite with nonfiction books
b)use of as is incorrect , when nouns are compared we need like
c)same as a and b
d)their has no antecedent we need its for public

But does 'its' in E has a logical antecedent since the noun in question, 'public' is in possessive form

its is a possessive pronoun.

A possessive pronoun can refer to a possessive noun

Thus its has a clear public antecedent in public's.

No problems in E.

Might be easier to see the comparison when you rephrase it to a common GMAT form:

"Like its interest in nonfiction books , the public's appetite for documentary films..."

Keep in mind that the extra words that emphasize the parallelism are also there to prevent ambiguity. The ambiguous sentence:

Bob likes Carol more than Alice could mean either Bob likes Carol more than he likes Alice (this is probably most peoples' default understanding) or Bob likes Carol more than Alice does. In spoken English, context often makes it clear (consider Bob likes hamburgers more than Alice; in this case, most peoples' default understanding would be Bob likes hamburgers more than Alice does rather than Bob likes hamburgers more than he likes Alice)

As thephoenix said, only similar items can be compared on the GMAT, and uwhusky's tip is a great way to test many sentences that just don't sound right: reorder the sentence to a more familiar form.