The gemara says that we learn out from the pasuk that prisha is necessary for both para aduma and yom kippur. The gemara explains that since the Torah uses both the term לעשות and לכפר, it can't just be referring to the para aduma because - פרה לאו בת כפרה היא, parah aduma isn't for a kapara, it is for tahara.
The Sha'agas Aryeh (in his sefer Gevuros Ari) asks based on the Midrash quoted by Rashi in Parshas Chukas, the Parah Aduma offers a kapara for the sin of the golden calf. Rashi cites the famous mashal of the בן שפחה who dirties the king's palace and the mother (parah aduma) needs to come to clean it up. Also, the gemara says in Moed Kattan - למה נסמכה פרשת מרים לפרשת פרה אדומה, לומר לך מה פרה אדומה מכפרת, אף מיתת צדיקים מכפרת. We see clearly that the parah aduma is coming for kapara, why does the gemara say that it isn't for kapara?
The Gevuros Ari answers that it was only the Parah Aduma done by Elazar in the time of Moshe that was serving as a kapara for the golden calf, but the parah aduma that was done in the future was not for kapara. Although the Gevuros Ari maintains this position, he points out that it still doesn't answer the question because maybe the pasuk of לכפר demanding prisha for the kohen gadol is specifically for parah aduma in the time of Moshe. Therefore, the Gevuros Ari concludes that although there is kapara offered through the parah aduma, the main function of the para aduma is for tahara and not to serve as a kapara, unlike Yom Kippur that the very essence of the day is for kapara. It was only אגב גררא מקופיא, that it also came to serve as a kapara for the eigel.
Regarding the first point of the Gevuros Ari that only the Parah Aduma of Moshe served as a kapara for the eigel, not the future parah adumos, it seems to be against Rashi in Chumash. After Rashi explains all the acts of the parah aduma and how they relate back to the eigel, he explains the pasuk (9) - והיתה לעדת בני ישראל למשמרת, that the concept of a mishmeres is that just as the sin of the eigel lingers throughout the generations, so too the parah aduma is למשמרת to linger for future generations. Simply this means that the parah aduma was not a one time thing, rather it was repeated a few time throughout the generations serving each time as a mishmeres for the חטא העגל. Perhaps the Gevuros Ari would interpret that only the ashes of Moshe's parah aduma lingered for many generations and was used for tahara for a very long time.
Regarding the conclusion of the Gevuros Ari that the parah aduma was not primarily for a kapara, rather it was done for tahara. It seems that the idea of it being a kapara is by virtue of the fact that it is being used for tahara. Meaning, the eigel brought tu'mah into the Jewish people, and the function of the parah aduma is to bring about tahara, the opposite of tu'mah. By bringing tahara it is indirectly serving as a kapara to fix the problem caused by the eigel.
The Shulchan Aruch (685) holds that reading the parsha of para aduma is d'oraysa, to which the Magen Avrohom says that he doesn't know what the source in the Torah would be. The Malbim in his sefer Artzos HaChaim says that since we have a mitzvah of zachor to remember how we angered Hashem in the midbar, and the main angering was the חטא העגל, we have an obligation to read the parsha of the eigel. However, since it is degrading to publicly read about the sin of the eigel, we achieve this by covering it up a little and read the parsha of para aduma which is a kapara for the eigel. The Meshech Chochma (Chukas) writes that since within the context of the miluim there is a reference to parah aduma, לעשות אלו מעשה פרה, and there is an opinion that by the miluim the reading of the parsha was me'akeiv (5b), so too by parah aduma the reading of the parsha is essential. According to this it would come out that the Torah reading of Achrei Mos which is the seder of Yom Kippur would also be d'oraysa.