Divided Brains & Divided Minds

In my Metaphysics textbook, there is an article by David Parfit describing split brain experiments and its implications for the philosophy of mind. So, the experiments go something like this. You take a brain and you split it into the two hemispheres. Each part of the brain controls a different part of the body. Next, the patient is shown a card that is half blue and half red. Now, say the right eye is focused on the red, and the left eye is focused on the blue. Then ask the patient which color he sees, and his right arm will write red, and his left arm will write blue, but the patient cannot comprehend both at the same time. He thinks he only sees blue and he also thinks he only sees red. He cannot conjoin the two beliefs. 

What Parfit says this proves is that our conscious is not a unified thing. Consciousness is supposed to  unify all beliefs, and this unified consciousness is what we call the self. But since we know that a single patient can have two streams of consciousness that cannot interact with one another, he is not a unified self. Parfit thinks this supports Bundle Theory. 

My initial reaction is that a unified self does not need to access all beliefs at once. Sometimes, the unified self forgets a thing and cannot summon some proposition with the rest of his current mental states. But that inability to have two mental states simultaneously doesn't show that his self isn't unified, and so split brain cases don't show that our inability to comprehend two things, or two sensations, at once that we are not a unified self. Further, I think it is too quick to identify our consciousness with the self. Parfit considers this but asks what reason is there to believe this? Why believe there are streams of consciousness that are not persons? One reason to believe this is to say that consciousness is a power of the self, but not identical to the self. And having a power means I am distinct from it. In this view, I am by nature the same self from when I was conceived, but I wasn't conscious at that moment. So, consciousness is a proper accident of a person, but isn't a person itself.