This is a little dated but worth looking at:

"the link between gluten sensitivity and problems with brain function, including learning disabilities, and even memory problems, is not that difficult to understand. Gluten sensitivity is caused by elevated levels of antibodies against a component of gluten called gliadin. This antibody (the antigliadin antibody) combines with gliadin when a person is exposed to any gluten-containing food like wheat, barley or rye.

When this happens, protein-specific genes are turned on in a special type of immune cell in the body. When these genes are turned on, inflammatory chemicals called cytokines are created. Cytokines, which are the chemical mediators of inflammation, are directly detrimental to brain function. In fact, elevated levels of cytokines are seen in such devastating conditions as Alzheimer’s disease, Parkinson’s disease, multiple sclerosis, and even autism. Essentially, the brain does not like inflammation and responds quite negatively to the presence of cytokines."
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You may recall Ron Hutton's experience with that abscessed tooth a couple of years back and how the cytokines from it knocked him for such a loop. Essentially, those of us with this facet of PD are maintaining a chronic state that is very similar so long as we eat wheat and the related grains.