Usage Of While

Dear Himanshu,
As per your pm, I'll add my two cents.

I would say --- "while" is a subordinate conjunction, which properly is followed by a full [noun] + [verb] clause. If we were going to hold a more-correct-than-Shakespeare, more-correct-than-the-GMAT, grammar-Nazi kind of approach, we might insist that "while" only could be followed by a full [noun] + [verb] clause. Of course, if we set that restriction, we would be setting the bar so high that even the GMAT would be wrong sometimes (i.e. in that GMAT-prep sentence), so for GMAT purposes, this is not a useful standard ---- of course, "while" can also be followed by a participle. Think of it this way. When there's a participle after "while", it's as if there is an implied subject and verb, an unspoken "they are" ----
A recent poll indicates that many people in the United States hold a combination of conservative and liberal political views; i.e., they denounce big government, saying government is doing too much and has become too powerful, while at the same time [they are] supporting many specific government programs for health care, education, and the environment. Of course, that changes the participle into a present progressive verb ---- we could change that into the simple present "they support". In some abstract sense, it's not "really" a participle there, and therefore, does not have the possibility of operating as a noun modifier. The entire "while" clause is an adverbial clause that modifies the verb of the main clause, "denounce", the verb to which the "while" clause provides contrast.
For more on adverbial clauses, see:
https://magoosh.com/gmat/2012/gmat-gramm ... d-clauses/
What KyleWiddison said is quite valuable. The contrast is between "denounce" and "supporting" ---- obviously, this is not grammatical parallelism, because they are in different forms, full verb vs. participle, but in some sense we could call this a kind of "logical parallelism", separate from grammatical parallelism. Those two words have a logical link that forms the backbone of the sentence, so of course they have the same subject --- the folks who "denounce" are the same as the folks who are doing the "supporting" --- that's the deep point of the sentence.

Does all this make sense?
Mike