Gemara

Rav: If a Zar slaughtered a Parah Adumah, it is Pasul. In the Parashah it says "Elazar" (who was the Segan) and "Chukah" (which teaches that it is Me'akev). However Shechitas Par Kohen Gadol of Yom Kipur by a Zar is Kosher.

Inference: Rav should also disqualify Shechitas Zar of Par Yom Kipur, for in that Parashah it says "Aharon" and "Chukah"!

Rejection: Shechitah is different, for it is not an Avodah.

Question: If so, also Shechitas Zar of a Parah Adumah should be Kosher!

Answer: A Parah Adumah is different. It is Kodshei Bedek ha'Bayis.

Question: If Bedek ha'Bayis requires a Kohen, all the more so Hekdesh Mizbe'ach does!

Answer (Rav Shisha brei d'Rav Idi): Shechitas Parah is like ruling about Tzara'as. Even though it is not Avodah, it requires a Kohen.

Shmuel disqualifies Shechitas Zar of the Par, for it says "Aharon" and "Chukah". He holds that Shechitas Zar of a Parah Adumah is Kosher.

Question: He should also disqualify regarding the Parah, for it says "Elazar " and "Chukah"!

Answer: Regarding the Parah it says "V'Shochat Osah Lefanav". This allows the Kohen to oversee while a Zar slaughters.

Rav expounds this verse to forbid Hesech ha'Da'as.

Support (for Rav - R. Yehoshua bar Aba - Beraisa) Question: Haza'ah of Mei Chatas is Kosher only through men and during the day. What is the source that also Shechitah, Kabalas Dam, Haza'as Dam, burning it, and throwing in the cedar, hyssop and Shni Tola'as must be during the day?

Answer: "Torah".

Suggestion: Perhaps we include even gathering the ashes, filling water (from a stream) and Kidush (putting the ashes on the water)?

Rejection: "Zos".

Question: Why do we include the former and exclude the latter, and not vice-versa?

Answer: We learn from Haza'ah of Mei Chatas. Just like it is Kosher only through men and only during the day, also Shechitah, Kabalas Dam, Haza'as Dam, burning it, and throwing in the wood, which are Kosher only through men, are Kosher only during the day;

We exclude gathering the ashes, filling water and Kidush, for women may do these.

Question: How does this support Rav?

Suggestion: Just like a woman may not slaughter, a Zar may not.

Rejection: A woman may not do Haza'ah of Mei Chatas, but a Zar may!

Answer (Abaye): "Elazar" excludes a woman. Likewise, it excludes a Zar.

43b - A reciter of Beraisos: Any Shechitah of a Zar is Kosher, except for a Parah Adumah.

R. Yochanan: Do not teach this! There is no Shechitah of a Zar that is Pasul.

R. Yochanan did not accept even his Rebbi's opinion about this!

R. Yochanan: R. Shimon ben Yehotzadak says that any Shechitah of a Zar is Kosher, except for a Parah Adumah. I say that it is Kosher. There is no Shechitah of a Zar that is Pasul.

Parah (3:9): The Kohen stands in the east facing west. He slaughters...

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 4:17): Everything done with a Parah Adumah from the beginning to the end must be during the day, and through male Kohanim.

Source (Kesef Mishneh): The Tosefta (Parah 4:3) says that everything done with a Parah Adumah must be through Kohanim. Some texts say 'through men'.

Rambam (ibid. 1:9,11,12): If one slaughtered a Parah Adumah to be Chulin it is redeemed...Even a Kohen Hedyot may burn a Parah Adumah; only the Parah in the Midbar required Elazar (the Segan). One who does a Parah Adumah wears the four Begadim of a Hedyot, whether he is a Kohen Hedyot or Gadol.

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

Answer #1 (Kesef Mishneh): The Sugya (Yoma 42b-43a) is better according to the opinion that allows a Hedyot. Also, in Sifri four Tana'im allow a Hedyot, and only R. Meir requires a Kohen Gadol.

Answer #2 (Emek ha'Netziv on Sifri Chukas, DH R. Yosi (p.143)): The Amora'im argue only about whether or not a Zar is Kosher, but all allow a Kohen Hedyot.

Rambam (Pesulei ha'Mukdashim 1:1,2): Anyone Pasul for Avodah may slaughter Kodshim, except for a Tamei; if he slaughtered, it is Kosher. Similarly, even though regarding Par Yom Kipur it says "V'Shochat Aharon", if a Zar slaughtered it is Kosher. Even a Parah Adumah is Kosher if a Zar slaughtered it. No Shechitah of a Zar is Pasul.

Source (Kesef Mishneh): The Rambam rules like R. Yochanan regarding Par and Parah, because in each case he agrees with Rav or Shmuel, and only one of Rav and Shmuel oppose him. Even though R. Yochanan's Rebbi disagreed, since R. Yochanan heard the opinion and rejected it we follow R. Yochanan.

Sefas Emes (42a DH Shechitas): Even though Rav (and R. Yochanan) say that Shechitas Parah by a Zar is Kosher, they allow l'Chatchilah.

Me'iri (42a DH Shechitas): L'Chatchilah, Shechitah of a Parah Adumah and Par Yom Kipur should be by a Kohen and the Kohen Gadol, respectively. B'Di'eved, both are Kosher even if a Zar slaughtered.

Likutei Halachos (Ein Mishpat 30): L'Chatchilah, R. Yochanan agrees with Rav who requires a Kohen for Shechitah. We learn from Par Yom Kipur, which (like all Avodas Yom Kipur) l'Chatchilah requires the Kohen Gadol. "Osah" permits a Kohen Hedyot for a Parah Adumah. 'We never find that Shechitah of a Zar is Pasul' is only b'Di'eved. The Rambam requires a Kohen l'Chatchilah.

Aruch ha'Shulchan ha'Asid (Kodshim 123:7): We cannot infer from 'Shechitah of a Zar is Kosher' that this is b'Di'eved, for this is Agav 'Shechitah of a Zar (of Par Yom Kipur) is Pasul'. We cannot make this inference from the Rambam's words, for he prefers to cite the Gemara verbatim. It appears that a Zar may slaughter l'Chatchilah, for Shmuel learns from "V'Shochat Osah Lefanav" - a Zar slaughters and the Kohen oversees.