The major signal that PARALLELISM was being tested here was the use of BOTH _______ AND _________. Whenever you see this structure, know that whatever comes directly after "both" must match what comes directly after "and."

We can break the sentence down like this:

...NFL Commissioner's Office has the responsibility both
1. to administer penalties unilaterally (for a wide range of infractions deemed harmful to the league and its image)
and
2. that they review their own decisions when subject to appeal from players.

The second item in this list must be an infinitive beginning with "to." That rules out everything except E.

To your question about whether a commissioner's office can review itself... the GMAT isn't always entirely consistent on that. With purely inanimate objects, we wouldn't want to use active verbs: "the house needs to protect itself against burglars" is illogical. We'd say "we need to protect our house against burglars."

But since offices / commissions / committees, etc., are made up of groups of people, the GMAT generally allows them to use active verbs (much like I did with "the GMAT allows" in this sentence). If the office can "administer penalties" in the non-underlined portion, we can hardly quibble with it reviewing itself in the underlined portion.

Does that help?

This two-part parallelism - also known as CLOSED PARALLELISM - is very common on the GMAT. Any time you see the following:
BOTH ____ AND ____
EITHER ____ OR ______
NOT ONLY _____BUT ALSO _______
NOT ________ BUT _________
etc,

... you need to make sure that what comes right after the first part structurally matches what comes after the 2nd part. For more, see:
https://www.beatthegmat.com/help-manhatt ... tml#713882
https://www.beatthegmat.com/sc-question- ... tml#769597
https://www.beatthegmat.com/spanish-arma ... tml#775935
https://www.beatthegmat.com/og-pg-725-81 ... tml#767953
https://www.beatthegmat.com/think-of-x-a ... tml#575578
https://www.beatthegmat.com/most-expendi ... tml#716319