A common misconception about the Parah Aduma

I am posting this again because every year I hear people make the same mistake.

We all know that the Parah Aduma is a chok and it is מטהר הטמאים ומטמא הטהורים. Most people think that this refers to the person who sprinkles the ashes, that the sprinkler becomes טמא. This would seem to be based on the simple reading of the pasuk (21) that says ומזה מי הנדה יכבס בגדיו. However, this is simply not true. Rashi there quotes the gemara that the מזה is actually טהור and the pasuk is coming to teach us that the person who carries it becomes טמא.