During binocular rivalry, conflicting images presented to the two eyes compete for perceptual dominance, but the neural basis of this competition is disputed. In interocular switch rivalry, rival images periodically exchanged between the two eyes generate one of two types of perceptual alternation: (1) a fast, regular alternation between the images that is time-locked to the stimulus switches and has been proposed to arise from competition at lower levels of the visual processing hierarchy or (2) a slow, irregular alternation spanning multiple stimulus switches that has been associated with higher levels of the visual system. The existence of these two types of perceptual alternation has been influential in establishing the view that rivalry may be resolved at multiple hierarchical levels of the visual system. We varied the spatial, temporal, and luminance properties of interocular switch rivalry gratings and found, instead, an association between fast, regular perceptual alternations and processing by the magnocellular stream and between slow, irregular alternations and processing by the parvocellular stream. The magnocellular and parvocellular streams are two early visual pathways that are specialized for the processing of motion and form, respectively. These results provide a new framework for understanding the neural substrates of binocular rivalry that emphasizes the importance of parallel visual processing streams, and not only hierarchical organization, in the perceptual resolution of ambiguities in the visual environment.