Gemara

Ula: "U'Nsatem Osah El Elazar ha'Kohen" - a Parah Adumah in the Midbar required Elazar, but for all generations it does not.

Version #1: In the Midbar Elazar was Kosher, but for all generations the Kohen Gadol is required.

Version #2: In the Midbar Elazar (the Segan) was required, but for all generations a Hedyot suffices.

Question: What is the source for the opinion that requires the Kohen Gadol for all generations?

Answer: We learn from a Gezerah Shavah "Chukah-Chukah" from Avodas Yom Kipur.

43a - Beraisa: "V'Chibes Begadav ha'Kohen" - in his (Bigdei) Kehunah.

Question: According to the opinion that allows a Hedyot, we understand why it must teach this. But according to the opinion that requires a Kohen Gadol, all the more so Bigdei Kehunah are required!

Answer: Indeed, a verse teaches this even though we could have learned from a Kal va'Chomer.

2a - Mishnah: Seven days before burning a Parah Adumah they would separate the Kohen who will burn it...

Parah (4:1): If Shechitah, Kabalah or Haza'ah of a Parah Adumah was Lo Lishmah it is Pasul...if the Kohen Gadol did not do it it is Pasul; R. Yehudah is Machshir. If he was Mechushar Begadim it is Pasul. It was done in the linen garments.

Rishonim

Rambam (Hilchos a Parah Adumah 1:11): Even a Kohen Hedyot may burn a Parah Adumah. Elazar burned the Parah in the Midbar even though Aharon was around. Subsequent Paros may be burned by the Kohen Gadol or a Hedyot.

Question: The Stam Tana of the Mishnah (Parah 4:1) requires the Kohen Gadol!

Note: In the Midbar, the Torah required Elazar for Haza'ah and burning. Surely, the Halachah is the same for both of these.

Answer #1 (Kesef Mishneh): Our Sugya leans towards the opinion that allows a Hedyot. Also, in Sifri four Tana'im allow a Hedyot, and only R. Meir requires the Kohen Gadol.

Objection (Emek ha'Netziv on Sifri Chukas, DH R. Yosi (p.143)): Elsewhere the Rambam rules like a Stam Mishnah even though it is an individual's opinion!

Answer #2 (Emek ha'Netziv, ibid.): The Amora'im argue only about whether or not a Zar is Kosher, but all allow a Kohen Hedyot.

Note: The Rambam (Parah Adumah 4:17) requires a Kohen for 'everything from the beginning to the end'. The Netziv understands that this includes Shechitah, therefore he does not distinguish Shechitah from other Avodos. Other Meforshim say that Shechitah is different. Amora'im argue about Shechitah of a Zar, but no one requires the Kohen Gadol. All require a Kohen to burn it; Tana'im argue about whether or not the Kohen Gadol is required.

Chazon Ish (Parah 8:4): The opinion that requires the Kohen Gadol for Parah for all generations learns from a Gezeirah Shavah "Chukah-Chukah"; from the Par of Yom Kipur. According to Rav, the opinion that requires a Kohen Gadol for burning and Haza'ah allows Shechitas Zar. Rav himself holds like R. Yehudah, who allows a Hedyot for burning and Haza'ah, and requires a Kohen for Shechitah.

The Chazon Ish (ibid.) first wrote that the Rambam requires a Kohen for Shechitas Parah, like Rav. The Chazon Ish retracted (Gilyonos b'Sof Parah DH Parah) because the Rambam (Pesulei ha'Mukdashim 1:2) says that Shechitas Zar is Kosher.

Rambam (ibid. 12): The one who does a Parah Adumah wears the four Begadim of a Hedyot, whether a Kohen Hedyot or Kohen Gadol does it.

Question (Mishneh l'Melech): The Gemara says that it is done in 'Bigdei Lavan', which connotes Bigdei Kohen Gadol of Yom Kipur; the Rambam rules that a Hedyot's Avnet contains wool. Also, the Sifri (Reish Chukas) learns Bigdei Lavan from "Chukah-Chukah" from Par Yom Kipur, which is done in the Bigdei Lavan of the Kohen Gadol. A Tosefta says that it was done in the Bigdei Lavan of a Hedyot, but this is not a good source, for the Tana could hold that a Hedyot's Avnet is all linen.

Answer (Emek ha'Netziv p.146 DH b'Kihuno) Question: The Rambam says that whether a Kohen Hedyot or Kohen Gadol does it, he wears the four Begadim of a Hedyot. But the Gemara requires the Kohen 'b'Kihuno', i.e. in Bigdei Kehunah. Bigdei Kohen Hedyot are not Kosher for the Kohen Gadol!

Answer: Rashi explains that 'b'Kihuno' means in Bigdei Kehunah. The Rambam must explain that it excludes other things that disqualify a Kohen, such as Shtuyei Yayin and one who did not wash his hands and feet. We learn from Mechusar Begadim that such people are not considered Kohanim regarding Avodah

Tosfos ha'Rosh (2a DH): The Mishnah says the Kohen who will burn a Parah Adumah. It is like the opinion that a Kohen Hedyot may burn it. Alternatively, it refers to the Kohen qualified to burn it.

Ritva (2a DH Shiv'as Yomim Kodem Sereifas): The Tana said 'Kohen' to avoid taking one side in the argument about whether or not a Kohen Hedyot may burn it. All agree that l'Chatchilah the Kohen Gadol burns it, and that every Parah (after Elazar's) was burned by a Kohen Gadol.