The opening phrase is intended to be grammatically equivalent to "the Motown Record Corporation." This means that it should be an appositive of "the Motown Record Corporation. If you are not familiar with term "appositive," you should look it up.
Essentially, "the Motown Record Corporation" = "one of the most successful African American businesses in history." In this sense, the opening phrase renames "the Motown Record Corporation.
Faulty parallelism occurs when two grammatical structures that ought to be identical to each other are not. A typical simple case would be: Jim likes hunting, fishing, and to swim. Here the forms "hunting" and "fishing" are parallel, but "to swim" is not the same type of construction, so it is not parallel.
Now, where I differ from the explanation that you were given is this: a participial phrase functions as an adjective, so it can modify a noun. The construction "Blah blah blah, noun verb whatever," will permit a participial phrase in the spot "Blah blah blah." For example, "Walking home from school, Sally spotted her dog running toward her down the sidewalk."
I think the reason that CB prefers "One" to "Being one" is that the Motown Record Company was not *already* one of the most successful African American businesses at the time that it was founded. The use of the present participle "being" tends to suggest that what happens in the main clause happens at the same time.
Another form of the sentence that would be ok is "Now one of the most successful . . . ," provided that the Mowtown Record Company still exists.
A form that would not be ok is "As one of the most successful . . .," because it wasn't the most successful when it was founded. It might have been founded in the hope that it would become one of the most successful, but it wasn't (technically) *founded* as one of the most successful.