The Gemara on Amud Aleph tells us that a woman may perform one aspect of the Parah Adumah ritual, the gathering of the ashes. It is not usually the case that women are involved in Temple service (although this was not technically a service in the fullest sense) so I thought it might be interesting to consider what is different about Parah Adumah.
There is a Midrashic teaching to explain the reason for the Parah Adumah. It is partially as an atonement for the Golden Calf. Rashi quotes a Midrash Tanchuma (Bamidbar 19:23):
פרה אדמה. מָשָׁל לְבֶן שִׁפְחָה שֶׁטִּנֵּף פָּלָטִין שֶׁל מֶלֶךְ, אָמְרוּ תָּבֹא אִמּוֹ וּתְקַנֵּחַ הַצּוֹאָה, כָּךְ תָּבֹא פָרָה וּתְכַפֵּר עַל הָעֵגֶל (תנחומא):
פרה אדמה A RED COW — Why this rite was performed with a cow may be exemplified by a parable it may be compared to the case of a handmaid’s child that defiled the king’s palace. They said: Let the mother come and clean up after her child. Similarly here: since they became defiled by the sin of the Golden Calf, let its mother (a cow) come and atone for the calf (cf. Midrash Tanchuma, Chukat 8).
Perhaps we can then say, that the involvement of women in the Parah Adumah ritual is related to the principle of אף הן היו באותו הנס, “They too, were involved in the miracle.” Where does this principle come up in Shas? It is used in regard to three halakhos: The Four Cups of Wine on Seder Night, The reading of the Megillah, and the lighting of Chanuka Candles. The simple explanation is that since women also experienced the redemption, so too they are obligated to rejoice and mark it through the rituals (Tosafos Pesachim 108b). However, Rashi and Rashbam (ibid) say a deeper peshat. It is not merely that they were involved in the experience of the miracle, rather they actually were essential to the miracle. Regarding the redemption from Egypt, the women were keeping families together so that the Jewish people existed in the first place. Regarding the reading of the Megillah, Queen Esther was a key player in the salvation. And, finally, regarding the Hasmonean victory, we attribute the story of Yehudis’ seducing and killing a general as part of the resistance.
In any case, though the principle of אף הן היו באותו הנס is regarding rabbinic mitzvos, perhaps this concept was learned out from the laws of Parah Adumah or applied to it. Thus, since it is symbolic of a mother cleaning up after her child, women’s potential involvement in the ritual was part of the observance.
Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation 
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