It seems to me that at this point in the text the phrase "parity reversal has to be a functor" is meant simply in the sense that "it is clear that parity reversal is a functor" ("has to be" in the sense of "it just has to be true").
Because for that to be true the operation given on superrings (the objects) needs to extend to an operation on the maps between them (the morphisms) and in this case it is clear that they do, because the maps preserve the grading and the operation simply swaps the gradings, so that the swapped gradings are then still preserved.
This is a common step when passing to category theory: At the beginning one typically does not so much introduce sophisticated category theoretic constructions, instead one goes along and recognizes that a lot of traditional constructions secretly are already category theoretic, and one just makes this explicit.
But one may still ask the question that you are asking, regarding the motivation to use tools from category theory. The answer is that looking at supergeometry via tools from category theory helps to organize concepts, avoid mistakes, and generalize appropriately.
- traditional approaches to supergeometry have difficulties with speaking about infinite-dimensional supermanifolds such as super mapping spaces, which are at the heart of physics. In the functorial perspective this generalization is immediate and transparent.
- traditional supergeometry has "principles" such as the one called "even rules", which are useful but remain a little mysterious. In the functorial perspective these principles gain the status of precise mathematical facts which leave no doubt about what is going on.