This is a statement from wikipedia. I don't understand what the "but not vice versa" part means. Does it mean that the composition of even or odd function with even is not odd (can only be even)? Or does it mean that the composition of even function with odd is not necessarily even? If it's the latter, then I don't understand how it works, please give examples?
First note that in general you should be careful with wikipedia articles on very elementary subjects, they are likely to be written by a low-level student with an imperfect understanding. So if something does not make a lot of sense, it may be just wrong.
Here I guess that the author wanted to say that even though "even $\circ$ even/odd = even", you don't have "odd $\circ$ even/odd = odd". Which is pretty obvious since the two are clearly incompatible.