I have never heard of a "fish-like parasite" but that doesn't mean anything because I'm not a parasitologist. It could be what you described or possibly a parasite that has broken up and parts have disintegrated??? Humaworm has often spoke about parasites looking different when they come out.
In my experience and research there is never just one parasite -- chances are if you have "one" there are many others as you have indicated.
After you have someone look into it for you, I'd be interested in knowing what it is. Please come back and post your findings. I look forward to always learning more.