The Sfas Emes (פרשת פרה תרמא) expounds on why we read פרשת פרה before פרשת החודש. Parshas Hachodesh is read in advance of the first month of the new year according to how the Torah counts the months. With the commencement of a new year, we are given a koach of התחדשות, renewal. We are granted an extra ability to recreate ourselves. We therefore need to cleanse ourselves first, in order to be able to utilize this gift properly. Through reading Parshas Parah, and cleansing ourselves, we become worthy vessels for the blessing of renewal. As Dovid Hamelech davens לֵב טָהוֹר בְּרָא לִי אֱלֹקִים וְרוּחַ נָכוֹן חַדֵּשׁ בְּקִרְבִּי (תהלים נא יב), create for me a pure heart, and a correct spirit renew within me. In order to merit a renewal of spirit, we must first cleanse our hearts, to create a vessel that can hold blessings.
Like we do teshuvah in Elul, before the beginning of a new year, so too, says the Sfas Emes, the time before the beginning of the year, according to the way in which the Torah counts the months, now in Adar, is an opportune time for teshuvah.