That line was by the person who wrote the article, Jeff Parsons, not the scientists. If you look at the press release, it says:

"It's not yet clear to us how the parasite generates energy," Prof. Huchon says. "It may be drawing it from the surrounding fish cells, or it may have a different type of respiration such as oxygen-free breathing, which typically characterizes anaerobic non-animal organisms."

So, they are not saying that these organisms don't use oxygen in some way, just that they are not oxygen breathers:

Some other organisms like fungi, amoebas or ciliate lineages in anaerobic environments have lost the ability to breathe over time. The new study demonstrates that the same can happen to an animal — possibly because the parasite happens to live in an anaerobic environment.