It was only after Katharine Graham became publisher of The Washington Post in 1963 that it moved into the first rank of American newspapers, and it was under her command that the paper won high praises for its unrelenting reporting of the Watergate scandal.
Option Elimination
(A) It was only after Katharine Graham became publisher of The Washington Post in 1963 that it moved into the first rank of American newspapers, and it was under her command that the paper won high praises -
Clause AND Clause - parallel
It was (It is a placeholder), and it was (it as a placeholder) - parallel
It was ....that it (refers back to Washington Post) and it was ....its (refers back to Washington Post) - parallel
(B) It was only after Katharine Graham's becoming publisher of The Washington Post in 1963 that it moved into the first rank of American newspapers, and under her command it had won high praises - "Katherine Graham's becoming" is wired "Katherine Graham BECAME" is better. The use of "had won" is incorrect. There is no other action in the past. Even if "becoming" is Became, it'll still not make sense - did it become popular even before she became a publisher - not the intended meaning.
Let me also share why "Katherine Graham's becoming" is wired.
In the English language, the possessive forms are typically used to show possession or ownership. In some instances, it can be used to indicate a relationship or association as well.
When we say "John's car," we mean the car that belongs to John. Here, when we say "Katherine Graham's becoming," - "becoming" is no possession or ownership by "Katherine Graham."
Let me share one more example if we need to use ING.
"The success of the company is attributed to Satya's leadership." Here, the possessive form "Satya's" is used correctly to show that the leadership belongs to Satya. However, if we were to use a gerund, we would say: "The success of the company is attributed to Satya becoming the leader." We wouldn't use the possessive form "Satya's" before "becoming" because it doesn't express possession or ownership in this context.
(C) Katharine Graham became publisher of The Washington Post in 1963, and only after that did it move into the first rank of American newspapers, having won high praise under her command - "that" is a pronoun, and here it is referred to an action of becoming a publisher - pronoun will refer to a noun and not an action. Wrong. Moreover, the usage of "having been" - the present perfect participle, is a wired here. Normally, the structure is "Having finished his work, John went for a walk" - the action of finishing work happened before John went for a walk. Present perfect participle is used to indicate an action that occurred before another past action (different from past perfect - in past perfect, the action has been completed in the past before another past action, but in present perfect participle, the effect of that action (that occurred before another past action) is present). Here, after "Having won," there is no other action.
(D) Moving into the first rank of American newspapers only after Katharine Graham became its publisher in 1963, The Washington Post, winning high praise under her command. - No verb. Just fragment.
(E) Moving into the first rank of American newspapers only after Katharine Graham's becoming its publisher in 1963, The Washington Post won high praise under her command - her can refer to Katherine Graham's - red herring. The use of "Katharine Graham's becoming" is wired, of course as explained above. Moreover, the modifier structure is a bit weird - Modifier 1 - Moving into the first rank of American newspapers, Modifier 2 - only after Katharine Graham's becoming its publisher in 1963 - both are modifying the Washington Post. Better construction would be - After KG became its publisher, The WP moved into the first rank of American newspapers and won high praise under her command for its unrelenting reporting of the Watergate scandal."