I've been looking at the idea that PD and inflammation are related and was surprised when I saw in wikipedia that "Neutropenia... is.. characterized by an abnormally low number of neutrophils" and "Neutrophils...serve as the primary defense against infections", which suggests there might be an inflammation-related link in your case.
I googled it a little more and found this study which I thought might be relevant http://www.ncbi.nlm.nih.gov/pubmed/10455900
. As you can see, in the results, it says that "Treatment with dopamine significantly increased phagocytic function". I'm not fully sure I'm reading this medical terminology accurately, but to me this means that when your dopamine levels go up then your Neutrophils also go up... and given that PD is characterized by low dopamine, it seems reasonable to presume that there is indeed a link between PD and low neutrophils (though it doesn't tell you what kind of link nor which side causes the other).
The same study also says that "Fenoldopam did not induce PMN apoptosis." Again, the medical terminology is not fully clear to me, but Feoldopman seems to be a dopamine agonist like the meds you're taking. If this is true, then this would mean that while your meds are useful for your PD symptoms, they might not have much of an impact in fixing your low neutrophils problem.
In my reading on PD and inflammation, I keep coming across "cortisol", an hormone in our bodies that (among other things) plays a key role in controlling inflammation. I also found a study ( http://www.ncbi.nlm.nih.gov/pubmed/2056538
) which suggests that cortisol levels correlate directly with neutrophis levels, and I've seen other studies suggesting that (a) dopamine and cortisol levels are correlated and (b) l-dopa meds (i.e., the ones most PwP end up taking sooner or later) actually reduce cortisol levels.
So, long story short, I think there are fair grounds to presume that Neutropenia, PD, and cortisol problems are linked. I'm not really sure what can be done about it, as none of the 3 seem easy to "fix", but personally, I'm trying to understand what to do with my (free) cortisol levels (which are lower than normal) just in case there's something in there. So, my suggestion for you is: (a) discuss these possible links with your doctor, (b) suggest they check your cortisol and transcortin levels - note that in my case my free cortisol levels are low just because I have too high transcortin levels; also note that checking this stuff can be done through either a simple blood and/or urine analyses; and (c) if you find that your cortisol and/or transcortin levels are out-of-norm, then discuss with your doctor if you can/should do something about it.
Hope this helps a little, and please do let me/us know how it goes,