Although we know that the entire Yom Kippur service is performed by the Kohen Gadol, nevertheless, Rav believes that he need not be the one who slaughters his par (the bull that is brought as a sacrifice by the kohen gadol). Rav explains that shehitah is not actually considered avodah (service), thus it can be done by anyone, even someone who is not a kohen.
At the same time, Rav rules that a para aduma – the red heifer that is burned and whose ashes are used as part of the purification process for someone who came into contact with a dead body (see Bamidbar 19) – can only be slaughtered by a kohen. This is true, even though a parah adumah is not considered to be a korban at all. It is not slaughtered in the precincts of the Temple, rather on the Mount of Olives, and is considered kodshei bedek ha-bayit – holiness that stems from its use for the Temple, not an actual sacrifice.
Rav Shisha, son of Rav Idi, said: The slaughter of the red heifer by a non-priest is invalid. The is just as in the case of appearances of leprosy, which, despite their not being a sacrificial service, still require the priesthood. Only a priest may declare the signs of leprosy to be pure or impure. It is apparent from this case that the logic of the a fortiori inference does not hold.
The Torah describes Biblical leprosy as a condition that can only be evaluated by a kohen. When a person sees a mark on his body that he suspects might be a nega tzara’at – a sign of leprosy – he shows it to a kohen who decides whether the nega should be disregarded, kept watch on, or declared to be leprosy. Even in generations where kohanim were not expert in evaluating the nega’im, they played an essential role. The trained Rabbi who examined the spot would offer his opinion about whether the nega was in fact leprosy, or not. In any case, the person remained tahor (ritually pure) until such time as the kohen – basing himself on the recommendation of the Rabbi – would declare the individual to be ritually impure.
Based on this we can conclude that there are halakhot that are the unique purview of the kohen, even though they are not connected directly with the Temple service. Rav suggests that para aduma is a similar case.