I think your biggest problem in the whole of this is microgravity. You can use vacutainer collection for blood, but unless a machine is doing the phlebotomy, I would assume it hard for the phlebotomist to draw from anyone that is an especially hard stick.
A lot of parasitology is recognized through stool specimens. How would one be able to collect said specimen? As far as I am aware, going to the bathroom on the ISS is no picnic. You could do a colonoscopy, looking for worms, but again, due to microgravity, that might be interesting. I do not know the protocol for laxatives in space.
I am a laboratory technologist by trade, and a lot of what we do is based on gravity. Immunology, Chemistry, Hematology, Urinalysis, Microbiology and even Blood Bank all deal with some sort of aliquoting. I know there is an analyzer on the ISS for some blood work, but it is limited. You would have to implement a lot of vacuum based technology, or something completely airless. In the second case, you might dilute specimens, and in the first case, a lot of your weight would go to pumps. Weight is a premium in getting things into space. With 3d printers, I can see some of it being solved (maybe a 3d printed set of channels to run specimen through so you won't have to worry about tubing) but I think technologically, we are still too far away for some of that stuff.
Also, in the case of parasitology, you would need to deal with microscopy...some of those buggers are small (i.e. plasmodium) and who knows what space might give us. We would have to figure out a way to deal with staining specimens so that artifact could be found.
I guess I could see a proactive approach just being someone in quarantine. Have a full alcohol bath to destroy most fauna (probably a towelette based system, or oversoaked towels). Afterward, have a robot run a phlebotomy procedure to extract blood to be put on a slide. Check a slide under the microscope for artifact. Lastly, do an ultrasound on the bowels to check for worms and the like.