Dedicated to the zchus of my Skype-Zevachim chavrusa who inspired this post.
The gemara in Zevachim [14b] tries to determine if shechting the Parah Aduma is an avoda. The Gemara concludes that it's not an avodah but nevertheless there is a גזירת הכתוב that it must be done by a Kohen מידי דהוה אמראות נגעים – just as we find that looking at tzaraas requires a Kohen even though it is not an avoda.
According to the Gemara's conclusion it would seem that shechting the Parah wouldn't require Bigdei Kehuna or Kiddush Yadayim Vi-raglayim that is usually required for avoda.
Another nafka minah would be based on the Rambam in hilchos Tumas Tzaraas [9/5] who says that a Chalalim are pasul from seeing tzaraas but Baalei Mumin are kosher. This would mean that Baalei Mumin may shecht the Parah Aduma whereas Chlalalim may not. This is the exact opposite of dinei avoda where a chalal's avoda is kosher [bi-dieved] while a Baal Mum's avoda is pasul. [See Shu"t Beis Yitzchak O"C 57/4, Chazon Ish Parah 8/3, Kehillos Yaakov Menachos 1 and Mishmar Halevi Zevachim 14].
However, the Tosfos Yeshanim [Yoma 42] and Tosfos HaRosh say that according to the opinion that we require a Kohen for the shechita of the Parah, Bigdei Kehuna WOULD be required as for other avodos. The Gevuros Ari adds that Kiddush Yadaim Vi-raglayim would also apply. According to them, the comparison the Gemara makes between Maros Negaim and the shechting of the Parah is not absolute because Maros Negaim require no Bigdei Kehuna and the like while shechitas parah does.
This opinion must be explained. Since the Gemara says that it is not an avoda and only necessitates a Kohen because of a גזירת הכתוב, why would Bigdei Kehuna be required?
The answer must be that the gemara in Yoma  teaches that the Kohen who engages in the process of preparing the Parah Aduma must wash his Bigdei Kehuna because they are tamei. So we learn that עשיית פרה אדומה requires Bigdei Kehuna. According to this, the גזירת הכתוב that שחיטת פרה אדומה requires kehuna teaches us that the שחיטה of the Parah is part and parcel of the עשיית פרה and therefore similarly requires Bigdei Kehuna as does the עשייה. Bigdei Kehuna are necessary NOT because the עשיית פרה is an avoda because it is not but because of a special גזירת הכתוב regarding the Parah. The Brisker Rov explains that for this very reason an אונן and מחוסר כיפורים are kosher for עשיית הפרה [Zevachim 17]. If it were an avoda then they would be pasul – but it isn't [זה לפי רש"י שם אבל עיין בתוספות]. A tamei person is pasul for עשיית הפרה but that is not because it is an avoda but because of a special גזירת הכתוב that a tamei may not be involved in עשיית הפרה. Please see Chiddushei HaGriz Hil. Parah Aduma page 62.
Based on this a perplexing Rambam shines brightly. The Rambam [פרה א' י"ב] writes that the עשייה of the Parah Aduma may performed by any kohen, even the Kohen Gadol, while wearing Bigdei Kohen Hedyot. How can this be??? If the Kohen Gadol is not wearing his regular clothing then he is מחוסר בגדים and his avoda is pasul.
We must protest!!!
We will publicly burn copies of Richard Dawkins books just for fun as a show of protest!!!
But based on the foregoing we understand very well that since it is not an avoda – the regular rule of wearing Bigdei Kohen Gadol is suspended and the clothing of a regular kohen are worn.
[ע' שיעורי רבי יחיאל מיכל זבחים י"ד, זהב מרדכי פ' חקת, ספר פרת חטאת עמ' רמ"ה, ס' דברות אריאל עמ"ס יומא סי' כ"ח שכולם צעדו בדרך זו אבל עי' בס' מנחת יעקב עמ' רל"א שרצה לומר פשט אחר ברמב"ם].
The great Rav Dovid Rapaport Hy"d ztz"l klers [Spanish for "asks"] if there is a din of יתור בגדים when involved in עשיית הפרה. He explains that we have a pasuk that מחוסר בגדים is pasul but that doesn't tell us anything about יתור בגדים since we are not dealing with a classical avoda where all of the normal laws of avoda apply. That shtims [Greek for "is consistent"] beautifully with what we have been saying.
The gemara [Zevachim 102a] records a machlokes if Moshe Rabbeinu was a Kohen Gadol. According to the opinion that he was, it would seem that he would also be a valid candidate for the shechting of the Parah as any other Kohen Gadol. However, the gemara [ibid] says that Moshe was pasul for Maros Negaim. According to that, HaGaon Rav Yitzchak Sorotzkin in his Rinas Yitzchak [Parshas Chukas], says if we accept an absolute parallel between the din of Maros Negaim and shechitas parah, it would emerge that Moshe was similarly pasul for shechitas parah.