Rashi, Ramban (both citing Yoma, 67b) and R. Bachye #1: The Satan and the nations of the world ask 'What sort of Mitzvah is this? What is the reason for it?' Therefore Hashem calls it a Chukah - a statute, as if to say 'It is My decree, and you do not have the authority to query it!'

R. Bachye #2: It is a Lashon of ?Chakikah? - carving; It is something that is carved in Heaven and hinted to Yisrael in the Parah Adumah. ¹

R. Bachye #3: It means a ?border? ² - to teach us to refrain from thinking about Chukim beyond what the Torah tells us due to their profundity. ³

R. Bachye #4: Because it is the only Chok which Sh?lomoh could not explain. ^{4 5}

R. Bachye #5 (according to Kabalah): Because it is brought outside the Beis-Hamikdash - to Midas Gevurah (the Midah of Yitzchak). ⁶

Menachos 27a: To teach us that the three sprinklings listed in Pasuk 6) and the seven Haza'os are crucial, ⁷ (and that, if Elazar sprinkles the blood she'Lo li'Shemo or not facing the entrance to the Heichal, it is Pasul - Rashi, in Menachos, 27b). ⁸

Sifri: To learn via a Gezeirah Shavah from "Vehaysah Zos lachem le'Chukas Olam" (Acharei Mos, Vayikra, 16:29) in connection with the Avodah on Yom Kipur, that the Avodah of the Parah Aduamah requires the four white Bigdei Kohen. ⁹

Ramban: Because, apart from the Sa'ir ha'Mishtale'ach, it is the only Korban to be brought outside the Mishkan.

R. Bachye #1 and Seforno (citing Yoma, 67b): Because at one and the same time, it is Metaher those who are Tamei, and Metamei those who are Tahor. ^{1 2 3}

R. Bachye #2 (citing the Tanchuma): Because a small Baheres the size of half a large bean renders a person Tamei, yet if it spreads to the whole body he is Tamei. ⁴

R. Sa'adya Ga'on: Because it is incomprehensible as to why its ashes are Metamei - bearing in mind that any other Tum'ah, once it turns into ashes, becomes Tahor. ⁵

Rashbam: It is the statute of Tum'as Meis (on Pasuk 14) - "Zos ha'Torah Adam ki Yamus be'Ohel".

Or ha'Chayim and Oznayim la'Torah: Because the Dini of Parah Adumah are due to the fact that Yisrael received the Torah at Har Sinai (which sanctified them; and Tum'as Meis is the result of the spirits who therefore crave to cling to the body of a Yisrael ¹ even after death, but not to a Nochri ² - Or ha'Chayim).

Yoma, 42b: To teach us that the following Avodos may only be performed by day ³ - Shechitah, Kabalas and Haza'as ha'Dam, ⁴ Sereifah and throwing into it the cedar branch, the hyssop-twig and the crimson, woolen thread.

Oznayim la'Torah: At Marah, Hashem gave them one of each of the three categories of Mitzvah - 'Muskalos' (self-understood) ... Dinim; 'Mekubalos' (logical but which are not automatically understood) ... Shabbos and Kibud Av va'Ein ⁵ and ; 'Chukim' (which defy logic) ... Parah Adumah, which is most profound the 'Chok of the Torah' - the most profound Chok of all, in that it contains contradictions (Metaher es ha'Teme'im and Metamei es ha'Tehorim),

Yoma, 42b: It precludes ¹ gathering the ashes, filling the water and the mixing from the obligation of performing the earlier Avodos by day. ²

Yoma, 2a: To teach us, via a Gezeirah Shavah from Vayikra, 8:34 "asher Tzivah Hashem La'asos" - in connecton with the Kohen Gadol on Yom Kipur - that the Kohen must separate from his wife seven days before burning the cow. ¹

Menachos 27a: It teaches us that the three speies (the cedar-wood, the hyssop and the crimson thread), the seven sprinklings are all crucial.

Chulin 24a: To teach us that the Shechitah is crucial, and tha if one kills it by breaking cutting the back of its neck, it is Pasul. ¹

Sifri: To teach us via a Gazeirah Shavah from Vayikra 16:29 - "Vehaysah Zos lachem le?Chukas OLam" - in connection with the Avodah on Yom Kipur - that the Avodah of the Parah Adumah must be performed wearing the four Bigdei Kehunah.

Ramban (on Sh?mos) and R. Bachye: Because the Torah sometimes expresses itself in this way. ¹ :

Rashi #1: It implies that the current Parah Adumah will always be named after Moshe - They will call it 'the Parah that Moshe prepared in the desert'. ¹

Rashi #2 (on Pasuk 22, citing R. Moshe ha'Darshan) and R. Bachye (citing a Midrash), TargumYonasan and Sifri: "Ve'yikchu" implies that Yisrael should purchase it with their money ² (which was taken from the T'rumas ha'Lishkah ³ [and "Eilecha", that Moshe should be the Gizbar {the financial supervisor} - Sifri]).

Ramban and R. Bachye: Initially, Hashem instructed Moshe to prepare a Parah Adumah just once in the desert. ⁴ Later - in Pasuk 10 - He extended it as a Mitzvah for future generations. ⁵

Moshav Zekenim and P'nei David 11, citing R. Efrayim: The Parshah alludes to Torah, which is called 'Lekach'. One who engages in Torah, it is called on his name. Parah is Torah, since it is P'rei ve'Revei (increases). One who reddens his face ⁶ for Torah, he retains it. Torah is Temimah, without any Mum (crookedness). Only one who engages in Torah is truly free (without a yoke). Elazar - one merits the crown of Torah only if Hashem helps) him (Ozer lo). ⁷ They elaborate at length).

Targum Yonasan and Sifri Zuta: A cow that is two years old ¹ (already from the time that it is purchased - Sifri Zuta). ²

Rashi and R. Bachye: It means ?completely red?. ^{1 2 3}

Rashi (on Pasuk 22, quoting R. Moshe ha'Darshan) and R. Bachye #1 (cting a Midrash): They sinned wih a calf; Let the mother serve as a Kaparah - This can be compared to the child of a Shifchah who dirtied the palace, and they said ?Let the mother come and clean-p the mess.

R. Bachye #2 (citing the Pesikta): "Parah" represents Mitzrayim, "Adumah" - Malchus Bavel; "Temimah" - Malchus Madai; "asher Ein bah Mum? - Malchus Yavan, and "ashr Lo Alah Alehah Ol" - Malchus Edom. ¹

R. Bachye: Nine: Moshe Rabeinu burnt 1; Ezra - 1; Shimon ha?Tzadik - 2; Yochanan Kohen Gadol - 2; Ali?ho?eini -I; hanamel ha?Mitzri - 1; Yishmael ben Fi?abi - 1. The tenth one will be burnt by King Mashi?ach, may it be speedily in our days.

Rashi (on Pasuk 22, quoting R. Moshe ha'Darshan) and R. Bachye (citing a Midrash): "Adumah" - because red symbolizes sin. Refer to 19:2:4:2*, ¹ and the Parah came to atone for the sin of the Eigel ha'Zahav; ² "Temimah" - because they were perfect (without blemish) and they became Ba?alei-Mumin - Le this cow come and atone for them so that they return to their state of perfection.

Moshav Zekenim: Parah Adumah is to atone for the Eigel ha'Zahav, which was made of gold, which is of a reddish color.

Moshav Zekenim and P'nei David 11: Refer to 19:2:4:3 and note.

Rashbam: It purified anyone who became Tamei Meis - whether by touching, carrying or Ohel ha'Meis.

Rashi #1: "Temimah" describes, not the cow, ¹ but the redness - to teach us that if it has two black hairs, it is Pasul. ²

Rashi #2 (in Pasuk 22, quoting R. Moshe ha'Darshan): Because before the sin of the Eigel they were perfect, and now they had become blemished 'Let the Parah come and atone for them, and they will return to their state of perfection'.

Sotah, 46a: It precludes the Eglah Arufah from becoming Pasul via a blemish. ¹

Targum Yonasan and Sotah, 46a: It incorporates working, ¹ having a yoke ² placed around its neck, being prodded with a spur or any other sharp object, or a male being intimate with it.

R. Bachye: Any blemish that disqualifies Kodshim, disqualifies a Parah Adumah.

R. Bachye: Because the Torah calls it a Chatas - in Pasuk 9.

Rashi #2 (on Pasuk 22, citing R. Moshe ha'Darshan): To atone for the fact that they removed from themselves the yoke of Heaven.

Ba'al ha'Turim: To teach us that the cow is Pasul even if one merely placed the yoke on it, without the cow actually drawing it.

Sotah, 46a: To invalidate it via a yoke even if it does not draw it, ¹ whereas other Melachos only invalidate it if it actually works. ²

Moshav Zekenim (based on Rashi on Pasuk 22): Since Parah Adumah atones for the Chet ha'Eigel, it is fitting that the mother comes and atones for her son. ¹

Moshav Zekenim #1: Sometimes the Torah writes what we could have learned from a Kal va'Chomer. ¹

Moshav Zekenim #2: One might have thought that just like Esnan (a harlot's wage), i.e. Ervah, does not disqualify it, since it is not offered in the Mikdash, also a blemish does not disqualify, since they are compared - "Mashchasam bahem Mum bam" (Vayikra 22:25). ²

Moshav Zekenim #3: The Torah writes "bah Mum" to extrapolte that a blemish does not disqualify the Eglah Arufah. One might have thought that it does, since Kaparah is written by it like by Kodshim. (The Halachah is, it disqualifies only Parah, which is called Chatas.)

Refer to 19:2:154:1.

Moshav Zekenim: We cannot learn it from a Kal va'Chomer since Korbanos must be offered by Kohanim who are Tahor, whereas a T'vul Yom is Kosher for Parah.

Moshav Zekenim #3: The Torah writes "Bah Mum", implying that a blemish does not disqualify the Eglah Arufah. One might have thought that it does, since Kaparah is written by it like by Kodshim.