They are two very close senses of the same word.
The second uses part in a countable noun sense that means "an amount, but not all, of a thing or a number of things".
The first uses part in an uncountable noun that means "some, a part or some parts [in the previous sense]". Because it is uncountable it is used with singular accord*
These two close senses amount to the same thing. "Parts of the company" emphasises the idea that these parts are separate from each other and hints that they may be identifiable, and you might favour it for that reason, while conversely you might favour "part of the company" precisely to avoid that suggestion; but they are just suggestions hinted at by the plurality rather than something explicitly stated, so either could indeed be used.
*Though note that when referring to a subgroup of a group of countable elements then plural accord is sometimes used with both, so "part of the soldiers were cut down", "part of the soldiers was cut down", "a part of the soldiers as cut down" and "a part of the soldiers were cut down" could all be found.