Two possibilities based on Ksav Sofer:
2) In a similar vein, the sin of avodah zarah is unique in that one is culpable even for the thought of committing idolatry. Even though those who actually worshipped the eigel were killed, the rest of Bnei Yisrael needed kapparah for thinking about it. Rashi writes with respect to parah adumah, “ain lecha reshus l’harhei achareha,” that even though it makes no sense, one is not allowed to question the process. (Does he mean this l’halacha, just like there is a halachic issur of thinking of committing idolatry? That would be a tremendous chiddush! I am more inclined to read it homiletically.)
I would like to suggest a different approach. The Midrash tells us that the parah adumah of Moshe remained (and still remains!) extant so that a little bit of its ashes could be mixed with subsequent paros. Chazal darshen on the pasuk, “V’Yikchu eilecha parah adumah…” that all paros are called the paros of Moshe; Moshe’s name is forever associated with the mitzvah of parah adumah (see Maharal in Gur Aryeh!).
Why was the cheit ha’eigel done? Because we gave up hope of seeing Moshe again. Rashi writes that the satan projected a vision of Moshe’s bier being transported to heaven. Moshe told us he would be back on a certain day; the time came and went (at least Klal Yisrael’s mistaken calculation of the time), and we felt abandoned by our leader. Parah adumah is the antidote to the cheit ha’eigel because it teaches us that Moshe Rabeinu will never abandon us – his parah will always be with us.
This helps answer another question as well. Why was the parsha of parah adumah placed here? The mitzvah of parah was given at Marah; the parah adumah had to have been used the first year after yetzi’as Mitzrayim to be metaheir those who needed it to bring korban Pesach. Perhaps the answer is that it is only post-mei meriva, when we knew with certainty that Moshe would not lead us into Eretz Yisrael to bring the ultimate geulah, that we needed the parah adumah to remind us that even though Moshe physically would pass on, his presence would remain with us for eternity.