The board of trustees, worried about the Neith corporation's rising debts, instructed Neith's CEO to not only downsize significantly, to lay off over 5000 employees around the country, but terminating as well the Neith's relationship with Arachne Unlimited, who in the past year were draining valuable funds from the Neith.
(A) to not only downsize significantly, to lay off over 5000 employees around the country, but terminating as well the Neith's relationship with Arachne Unlimited, who in the past year were
(B) to not only downsize significantly, to lay off over 5000 employees around the country, but also to terminate the Neith's relationship with Arachne Unlimited, which in the past year were
(C) not only to downsize significantly, to lay off over 5000 employees around the country, but also terminating the Neith's relationship with Arachne Unlimited, who in the past year was
(D) not only to downsize significantly, laying off over 5000 employees around the country, but also to terminate the Neith's relationship with Arachne Unlimited, which in the past year wavers
(E) not only to downsize significantly, laying off over 5000 employees around the country, but also terminating the Neith's relationship with Arachne Unlimited, who in the past year were

First, we should read the meaning of the sentence, "laying off over 5000 employees" modified the verb "to downsize significantly" and is not the same logical parallel item. Moreover, we scan through the parallel item with the marker "not only A, but also B", we will spot the correct one as choice D.
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1. The first reasoning is that Arachne Corp.’ is a non-human singular entity; so, we need- which was- D is the only choice with the combination.
2. Second split on correlative //ism; not only to… but also to. D is once again the only choice with that combination
I think two splits are good enough. But to know the tenets of grammar thoroughly

3. Modification; to downsize is through the process of laying off; so laying off needn’t be parallel to downsize.This leaves D and E. Between D and E, we can eliminate E either on correlative conjunction or on SV mismatch of the verb were.
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Vote for D, can any one help me with understanding sentence structure ???

The board of trustees, worried about the Neith corporation's rising debts, instructed Neith's CEO not only to downsize significantly, laying off over 5000 employees around the country, but also to terminate the Neith's relationship with Arachne Unlimited, which in the past year was draining valuable funds from the Neith.

Main Clause
The board of trustees, worried about the Neith corporation's rising debts, instructed Neith's CEO not only to downsize significantly, laying off over 5000 employees around the country, but also to terminate the Neith's relationship with Arachne Unlimited,

Subject: The board
Verb: instructed (not sure)

"not only.. but also" (coordinating conjunction)

Worried is verb-ed modifiers
laying - verb-ing modifer - modifies the clause "not only to downsize significantly"

Subordinate clause:
which in the past year was draining valuable funds from the Neith.
here "which" refers to "Arachne Unlimited"
subject: which
verb: was draining