The change in color to the skin (redness, stinging and white) is related to the astringent effect of the alcohol in the Purell. I cannot think of a black parasite that would behave as you describe. Scabies (a skin mite) tend not to be black however if you see a dermatologist then skin scraping will confirm the presence of mites if they are there.
I would recommend that you keep some adhesive tape handy, the next time you see a black speck, apply the tape to lift it of the skin then seal it on a sheet of white paper. That will allow the dermatologist to see it.
Lice are black, but large enough that you can see with the naked eye on body hair, eyebrows etc.
You would not see a bacteria or virus as these need microscopes to identify. Ther dermatologist wll also be able to determine what the sores are. I would recommend that you stop using the Purell as it is iritating to the skin when used regularly on areas that are softer thsn the hands.