Official Explanation
This one is tricky because it involves parallelism at a couple different levels. First of all, we are comparing Hannibal's vs. Scipio's approach. Hannibal's technique was "charging", so we suspect Scipio's counterstrategy would be parallel to this: "commanding", not "to command." This makes us suspicious of (A) & (B). If there were no parallelism to consider, the use of the infinitive in this context would be a perfectly correct construction.
After the word command/commanding, the most natural idiom is an infinitive:to command B to do X. The "that"-clause structure here, "that his men should part", is very awkward, and does not fit. Because of this, we reject (A) & (D).
The structure "with them killing" is 100% wrong. In general, the GMAT hates "with" + [noun] + [participle] to encapsulate action. If we want to talk about action, we need to use a full bona fide verb. This structure appears in both (A).
We know that Scipio is commanding his men to do something: what is he commanding them to do? He is commanding two things: (a) to part, and (b) to kill the elephants. These two actions need to be in parallel as well. Furthermore, the correction idiom is "command" + [infinitive], while the construction "command" + [gerund] is 100% wrong. Choice (C) correctly has two infinitives, "to part" and "to kill". Choice (C) is the only answer that has correct parallelism on both levels, so this is the only possible answer. (C) is correct.
Both (D) and (E) lose the important distinction between the two things that Scipio commanded: to part and to kill. Instead, (D) and (E) both create a list of participles and in doing so Scipio is commanding only that his men part, a major change in meaning. Remember, parallelism is not automatically correct especially when it entails a major change in meaning.