The answer is D.
There are two issues at play: Parallelism within a list and Modifiers
The intent of the sentence is to illustrate a list of things that Joan of Arc did.
Joan of Arc...
turned the tide of English victories... and
persuaded Charles to claim his throne.
Because the 1st item in the list is "turned", the last item must be "persuaded" but cannot be "pursuading"
The reason I can tell that the list is not
liberating the city and
persuading Charles
is that there is no answer that gives us a good grammatical structure for this list.
(C) would say "Joan turned the tide by
liberating the city and
persuading that the throne be claimed by Charles"
There is one problem: "persuading" should be followed by Charles ie. "persuading Charles to claim his throne."
(E) would say "Joan turned the tide by
liberating the city and
persuading that Charles should claim the throne"
This is no better than C. To be acceptable, it should say "persuading Charles to claim his throne."
Once we remove all answers that use "persuading", we are left with A, B and D.
(A) simplifies to "Joan...turned the tide of English victories...and
she persuaded Charles to claim his throne"
A is grammatically fine, but "she" is redundant so stylistically A is not the best choice.
(B) simplifies to "Joan...turned the tide of English victories...and persuaded Charles...in claiming his throne"
B commits a modifier error. "perusade X to do Y" is correct. "persuade X in doing Y" is incorrect.
(D) simplifies to "Joan...turned the tide of English victories...and persuaded Charles to claim his throne". Perfect!