korban pesach and parah adumah

The Midrash plays up parallels between parah adumah and korban pesach.  Shmos Rabbah 19 calls attention to both being called a chukah: 

 דָּבָר אַחֵר, יְהִי לִבִּי תָמִים בְּחֻקֶּיךָ, זֶה חֻקַּת הַפֶּסַח וְחֻקַּת פָּרָה אֲדֻמָּה, לָמָּה, שֶׁשְּׁנֵיהֶן דּוֹמִין זֶה לָזֶה, בָּזֶה נֶאֱמַר: זֹאת חֻקַּת הַפָּסַח, וּבָזֶה נֶאֱמַר (במדבר יט, ב): זֹאת חֻקַּת הַתּוֹרָה, וְאִי אַתָּה יוֹדֵעַ אֵיזוֹ חֻקָּה גְּדוֹלָה מִזּוֹ. מָשָׁל לִשְׁתֵּי מַטְרוֹנוֹת דּוֹמוֹת שֶׁהָיוּ מְהַלְּכוֹת שְׁתֵּיהֶן כְּאַחַת נִרְאוֹת שָׁווֹת, מִי גְדוֹלָה מִזּוֹ אוֹתָהּ שֶׁחֲבֶרְתָּהּ מְלַוָּה אוֹתָהּ עַד בֵּיתָהּ וְהוֹלֶכֶת אַחֲרֶיהָ. כָּךְ בַּפֶּסַח נֶאֱמַר בּוֹ: חֻקָּה, וּבְפָרָה נֶאֱמַר בָּהּ: חֻקָּה, וּמִי גְדוֹלָה, הַפָּרָה, שֶׁאוֹכְלֵי פֶּסַח צְרִיכִין לָהּ, שֶׁנֶּאֱמַר (במדבר יט, יז): וְלָקְחוּ לַטָּמֵא מֵעֲפַר שְׂרֵפַת הַחַטָּאת, הֱוֵי: כָּל בֶּן נֵכָר לֹא יֹאכַל בּוֹ.

Shmos Rabbah 17 notes the similarity between the sprinkling done with the azov bundle for korban pesach and the same done for parah adumah:

 בְּפָרָה אֲדֻמָּה צִוָּה לְהַשְׁלִיךְ בִּשְׂרֵפָתָהּ עֵץ אֶרֶז וְאֵזוֹב, וְהַזָּאַת מֵי נִדָּה צִוָּה לַעֲשׂוֹת בָּאֵזוֹב, וְטַהֲרַת הַמְצֹרָע צִוָּה לַעֲשׂוֹת בְּעֵץ אֶרֶז וְאֵזוֹב. וְכֵן בְּמִצְרַיִם צִוָּה לְהַגִּיעַ הַדָּם אֶל הַמַּשְׁקוֹף וְאֶל שְׁתֵּי הַמְּזוּזוֹת בָּאֵזוֹב, שֶׁנֶּאֱמַר: וּלְקַחְתֶּם אֲגֻדַּת אֵזוֹב

There seems to be a common thread running between these two mitzvos, an idea that Rav Bakshi Doron develops at length in an essay in his teshuvos.  

We have a din that ain kateigor naasah saneigor, e.g. the kohen gadol does not wear bigdei zahav when he does avodah lifnim on Y"K because gold is associated with cheit ha'eigel and we don't want that negative association hanging around our neck.  So why is it, asks the ShLa"H, that a sheep, the very animal that the Egyptians worshipped, is used for korban pesach?  Why use an animal that has a negative association with idolatry, which we ourselves were guilty of in Mitzrayim?  

Why in our parsha us Aharon told to offer an eigel when that is the very animal of the sin of cheit ha'eigel?  

The ShLaH answers that with respect to avodah on Y"K, the gold garments are incidental to the avodah taking place.  In that case, we don't want to introduce anything that has a cheit association into the mix.  However, with respect to korban pesach, the whole point of offering the korban is to undermine the avodah zarah.  The same with the eigel korban in our parsha, the whole point of the korban is to show that we no longer have a connection with cheit ha'eigel.  We davka want to bring up the association in order to negate it.

Rav Bakshi Doron suggests that this yesod of the ShLaH is behind the link between k"p and parah.  Parah adumah paradoxically is mtamei tehorim but mitaheir tmei'im.  So too, when we offer k"p, paradoxically the same animal that had been worshipped as avodah zarah becomes the vehicle by which we negate idolatry and come to avodas Hashem.  

The gemara likens sprinkling the blood of k"p on the doorposts to zerikas ha'dam of other korbanos, but based on the link to parah adumah, Rav Bakshi Doron suggests that just as parah is a m'taheir, so too the k"p served as a m'taheir for the homes where avodah zarah had previously been worshipped.