The parasites are from not cooking foods thoroughly. Crapping in a toilet or in the dirt isn’t gonna change that. The damned parasites neither prove nor disprove anything.
I do wonder about the baths, though. No chlorine. How often was that bath water changed during the day? Seems to me that public baths in that era served as no more than petri dishes.
Warm water, oils and other nutrients from skin and such like... mix in masses of bacteria from hordes of bathers...
http://www.impetigopictures.org
The Romans didn’t use valves very much, so all the baths had slow-in, slow-out water flow, iow, analogous to the way large buildings in the modern era change over the air. Baths which were not fed by aqueduct (like Aquae Solis, at modern Bath in England) were spring-fed, and the water cycled in, was used, and cycled out. The biggest problem would have been typhus, and there aren’t any known large outbreaks of that during Roman times. The plague in Athens during the Age of Pericles has been attributed to typhus, and it was a commonplace in the Nazis’ extermination camps.