Mishnah Parah 3:6: And they would make a ramp from the temple mount to the mount of anointing [the Mount of Olives], arches on top of arches, and [hollow] domes over the foundations, because of graves of the deep, for on it the priest who would burn the heifer, and the heifer and all its helpers would leave to the mount of anointing.

וְכֶבֶשׁ הָיוּ עוֹשִׂים מֵהַר הַבַּיִת לְהַר הַמִּשְׁחָה, כִּפִּין עַל גַּבֵּי כִפִּין, וְכִפָּה כְנֶגֶד הָאֹטֶם, מִפְּנֵי קֶבֶר הַתְּהוֹם, שֶׁבּוֹ כֹהֵן הַשּׂוֹרֵף אֶת הַפָּרָה, וּפָרָה וְכָל מְסַעֲדֶיהָ, יוֹצְאִין לְהַר הַמִּשְׁחָה:

Mishnah Parah 3:7: If the heifer did not want to leave, they could not lead it with a black [cow], lest [the public] say, "they slaughtered the black one." Nor [could they lead it] with a red one, lest they say, "they slaughtered two." Rabbi Yose says: Not for this reason, but rather because the verse says (Numbers 19:3), "And he shall take her out," [her] alone. And the elders of Israel would precede it by foot to the mount of anointing, and a house for ritual immersion was there. And they would make the priest who would burn the cow ritually impure because of the Saducees, so they would not say, "it was done by one for whom the sun had set." [There are various forms of ritual impurity for which to become totally pure, one must immerse in water and then wait until the next sunset. The Saducees believed that the priest who burnt the red cow needed to be totally pure such that he'd only be allowed to do it if he waited for sunset. The sages held that he is able to burn the cow, even after only immersing in water, thus they made him impure and had him immerse so that everyone could see that they were not following the opinion of the Saducees.]

לֹא הָיְתָה פָרָה רוֹצָה לָצֵאת, אֵין מוֹצִיאִין עִמָּהּ שְׁחוֹרָה, שֶׁלֹּא יֹאמְרוּ, שְׁחוֹרָה שָׁחֲטוּ. וְלֹא אֲדֻמָּה, שֶׁלֹּא יֹאמְרוּ, שְׁתַּיִם שָׁחֲטוּ. רַבִּי יוֹסֵי אוֹמֵר, לֹא מִשּׁוּם זֶה, אֶלָּא מִשּׁוּם שֶׁנֶּאֱמַר (במדבר יט), וְהוֹצִיא אֹתָהּ, לְבַדָּהּ. וְזִקְנֵי יִשְׂרָאֵל הָיוּ מַקְדִּימִים בְּרַגְלֵיהֶם לְהַר הַמִּשְׁחָה, וּבֵית טְבִילָה הָיָה שָׁם. וּמְטַמְּאִים הָיוּ אֶת הַכֹּהֵן הַשּׂוֹרֵף אֶת הַפָּרָה, מִפְּנֵי הַצְּדוֹקִים, שֶׁלֹּא יִהְיוּ אוֹמְרִים, בִּמְעֹרְבֵי שֶׁמֶשׁ הָיְתָה נַעֲשֵׂית: