Throughout the brain, parallel processing streams compose the building blocks of complex neural functions. One of the most salient models for studying the functional specialization of parallel visual streams in the primate brain is the lateral geniculate nucleus (LGN) of the dorsal thalamus, through which the parvocellular and magnocellular channels, On-center and Off-center channels, and ipsilateral and contralateral eye channels are maintained and provide the foundation for cortical processing. We examined three aspects of neural processing in these streams: (1) the relationship between extraclassical surround suppression, a widespread visual computation thought to represent a canonical neural computation, and the parallel channels of the LGN; (2) the magnitude of binocular interaction in the parallel streams; and (3) the magnitude of suppression elicited by perceptual competition (binocular rivalry) in each stream. Our results show that surround suppression is almost exclusive to Off channel cells; further, we found evidence for two different components of monocular surround suppression-an early-stage suppression exhibited by all magnocellular cells, and a late-stage suppression exhibited only by Off cells in both the parvocellular and magnocellular pathways. This finding indicates that stream-specific circuits contribute to surround suppression in the primate LGN and suggests a distinct role for suppression in the Off channel to the cortex. We also examined the responses of LGN neurons in alert macaque monkeys to determine whether neurons that supply the cortex with visual information are influenced by stimulation of both eyes. Our results demonstrate that LGN neurons are not influenced by stimulation of the non-dominant eye. This was the case when dichoptic stimuli were presented to classical receptive fields of neurons, extraclassical receptive fields of neurons, and when stimuli were appropriate to produce the perception of binocular rivalry.