What part of the brain controls depth perception?
According to the article: Scientists uncover second depth-perception method in the brain:


  the neurons in the middle temporal area of the brain are combining visual information and physical movement to extract depth information


This Scientific American article states that there is a process involved:


  Visual-image processing from the eye to the brain happens in stages. Rudimentary features such as the orientation of edges, direction of motion, color, and so on are extracted early on in areas called V1 and V2 before reaching the next stages in the visual-processing hierarchy for a progressively more refined analysis. This stage-by-stage description is a caricature; many pathways go “back” from stage to stage—allowing the brain to play a kind of 20-questions game to arrive at a solution after successive iterations.


This process is discussed further in this article with what is known as the Laminart model.