פירוש על קידושין 72:2

The sprinkling of the ashes of the red heifer can be performed only by the deputy high priest, as it was by Elazar in the wilderness. The sprinkling inside the Holy of Holies must be performed by the high priest. If even other priests cannot do this sprinkling, all the more so women cannot. Thus the mishnah should not even need to state this.

The sprinkling referred to is the sprinkling of the blood of a bird offering on the altar. That it must be offered by a male priest is derived from an inference from animal offerings. Even an Israelite may slaughter an animal offering, but only a priest can sprinkle its blood. So if only a priest can slaughter a bird offering, obviously only a priest can sprinkle its blood.