The Maharsha here discusses at length the significance of Yom Kipur, that it is "Shabbos Shabbason," making it even holier than Shabbos. It is more like Olam ha'Ba than Shabbos, he states, in that there is no eating or drinking, and there is no Satan (as he the Satan does not prosecute on Yom Kipur, only the rest of the year).

Although the Maharsha does not say so, this would obviously create a similar significance to accepting the Torah at Har Sinai, when Bnei Yisrael had their holiest day ever. This is when they ascended to the level of Adam ha'Rishon before the sin of the Etz ha'Da'as, when the Satan was external, not an internal part of them. Extending this theme, it is possible that this is also somewhat applicable to the Parah Adumah. The Para Adumah is an atonement for the sin of the Golden Calf (see Rashi in Bamidbar, 19:22 DH "Para"). This helps remove the greatest sin of Bnei Yisrael, which destroyed their holy stature after receiving the Torah. Accordingly, preparing to atone for this sin also involves showing that we wish to return to the holy status that we had when we received the Torah. It is therefore appropriate that the Kohen affecting this atonement prepare for it in the same manner for which Bnei Yisrael prepared for Matan Torah.

Kol Tuv,

Yaakov Montrose