In point of fact, the sentence is parallel. It's also an example parallelism allowing you to not have to repeat portions of the sentence. (This is also referred to as an elliptical sentence.)
The sentence happens to be ungrammatical because of its parallelism.
The parallel structure is this:
They should be X not only (about Y) but (about Z).
This expands into the following:
They should be X not only about Y, but they should be X about Z.
In the example sentence, this is wrong:
✘ 1. During times of change, they should be thinking not only about the strain on their staff but take time out to think of themselves.
It can be parsed in this way:
During times of change, they should be thinking not only (about the strain on their staff) but (take time out to think of themselves).
This expands into:
✘ During times of change, they should be thinking not only about the strain on their staff, but they should be thinking take time out to think of themselves.
In order for the second part of the sentence to work correctly with such a parallel structure, the sentence needs to be rephrased slightly so that not only occurs earlier:
✔ 2. During times of change, they should not only be thinking about the strain on their staff but take time out to think of themselves.
This results in the following parsing:
During times of change, they should not only (be thinking about the strain on their staff) but (take time out to think of themselves).
And this expands into:
✔ During times of change, they should not only be thinking about the strain on their staff, but they should take time out to think of themselves.
To summarize:
✘ 1. During times of change, they should be thinking not only about the strain on their staff but take time out to think of themselves.
✔ 2. During times of change, they should not only be thinking about the strain on their staff but take time out to think of themselves.
The essential change in the construction that allows the parallel construction to work is the change in location of not only:
they should be thinking not only VS
they should not only be thinking
In the original, be thinking applies to both items in the parallel structure because it comes before not only. (And results in the incorrect be thinking take time out.)
In the rephrased version, be thinking only applies to the first item.
There is also this comparison:
- They should not only be thinking, but they should take time out. OR
- They should not only be thinking, but they should be taking time out.
There is nothing ungrammatical about either version. Whether or not the second item should have the same verb tense as the first is entirely a matter of style.