I think Promoters of Glutathion (And I am not one of them) do not claim that it improves PD symptoms but rather slows down progression of disease. The sited research deals with improvement in symptoms and not neuroprotection .
Never the less, I remember reading some where that glutathion does not cross the Blood Brain Barrier so it will be useless any way. N-ACETYCYSTENE, a glutathion agonist which crosses BBB is taken by some PD patients (I am included) in the hope to boost glutathion in the brain.
Imad