I'm not a biologist so I'm not sure what a "free-living stage" is or what an "obligate" parasite is. Below is a write up on the life cycle of Brook that I can understand. What I draw from it is, the tank the fish came out of has to be fallow (fishless) for 4 weeks to make sure Brook is no longer in the tank. Most writer say 6 to 8 weeks. How long you leave it fishless is up to you. The below writeup is specifically for the Brook parasite
"LIFE CYCLE - Once reproduction has taken place, the newly formed protozoa is able to freely swim by the use of its ciliates and able to find a new host or more usually, attach to the same fish its "parents" are on, which is why you will note such a rapid progression of this parasite. Attachment usually occurs at the gills of the fish first and spreads out wards as they multiply, as this parasite feeds upon the fish, it also releases toxins that can prove fatal to the fish very quickly. Free swimming protozoa can persist for quite some time without a host to feed upon. As such, any aquarium that has had this parasite introduced to it, must remain fallow (fishless) for no less than four weeks in order for the remaining free swimming parasites to die off without a host to feed upon. As such, all fish within the aquarium must be put into a quarantine tank and treated, while being kept out of the main aquarium during that four week period."