Is this grammatically correct? I mean, it is alright for me to write who "one of them" is in an entirely new sentence? Is it considered a sentence fragment and therefore grammatically incorrect?

"By one look of his now unnaturally light pupils, she could tell that he had become one of them. A slave to the parasite that overrode his mind and controlled every movement of his body."

This is absolutely correct!

Yes, the sentence structure and usage in your example is grammatically correct. Starting the sentence with "A slave" creates a sentence fragment, but it is acceptable to use fragments for stylistic purposes, as is the case in your writing. It is a common technique used in creative writing to add emphasis and create impactful sentences. In this case, by starting the second sentence with "A slave," you draw attention to the fact that the protagonist has become controlled by the parasite. So, even though it is a fragment, the use is intentional and contributes to the overall effect of the passage.