Balancing a home delivery pizza with one hand and gripping a six-pack carton of Coke with another, the young boy Vijay paused in front of the guard room of the building.

The primary issue with the original sentence:  when referring to two hands, the correct phrase would be "with the other," not "with another." This is because "another" implies an additional, unspecified hand, which is illogical since humans only have two hands. Only Option D corrects this error. Furthermore, D simplifies and presents a more straightforward version of "a six-pack carton of Coke" by replacing it with "a pack of six Cokes."

Aside from the obvious issue with using "another," the remaining choices are rife with grammatical errors and unidiomatic phrases. Option B, for instance, can be eliminated because the phrase "having gripped" creates an unnecessary tense shift; "balancing" and "gripping" should be in the same continuous form for parallelism. Similarly, in Option C, the use of the phrase "having a balance of" is unnatural and awkward. 

Hence, Option D is the best choice.