Mayan civilization flourished at the same time as Greek

They're both correct. And sorry piyatiwari, but both of your examples are correct as well.

A comparison or other parallel structure that attributes exactly the same tensed verb to a couple of different nouns can (1) repeat the verb, (2) use a pro-verb, or (3) elide the verb. That yields three correct versions of piyatiwari's example sentence:

Jim jumped at the same time as Dwight jumped.
Jim jumped at the same time as Dwight did.
Jim jumped at the same time as Dwight.

The practical effect of this for GMAT SC is that you should look elsewhere in the sentence to eliminate answers.

ADDENDUM 1:
I guess that you saw the question below, but you didn't want to reproduce it here because it's from the OG. As it happens, the question is also in the GMAT PrepTest, so it's not banned.

Salt deposits and moisture threaten to destroy the Mohenjo-Daro excavation in Pakistan, the site of an ancient civilization that flourished at the same time as the civilizations in the Nile delta and the river valleys of Tigris and Euphrates.

A. that flourished at the same time as the civilizations
B. that had flourished at the same time as had the civilizations
C. that flourished at the same time those had
D. flourishing at the same time as those did
E. flourishing at the same time as those were

I'll let you look for an existing thread on the rest of the question, and just say now that you shouldn't worry about the difference between "at the same time as the civilizations" and "at the same time as the civilizations did," not least because you're not given that choice here.

ADDENDUM 2:
This is irrelevant to the GMAT, but may be interesting to some even so. Have you noticed that most people say "Jim jumped at the same time as me," but some people say (or more likely write) "Jim jumped at the same time as I"? The latter, arch though it sounds, is preferred by the sort of people who diagram sentences. "I," such people argue, is the subject of the elided verb "jumped." (I don't have a dog in this fight, except insofar as I mean to stubbornly defend the position I took in an argument with my mother when I was thirteen years old.)