The explanation above explains why parshas Parah precedes parshas Hachodesh (because Parah is “the purity of the Jewish people“), although the parah adumah (red cow) was offered on the second day of Nissan after Hachodesh (which refers to the first of Nissan). In the service of Hashem, Hacodesh refers to the service of the tzaddik (righteous), for at the time of the Exodus all the Jewish people were just like a convert who is considered to have been born at the moment of conversion. Parah refers to the service of teshuvah (repentance). That is why the portion of Hachodesh preceded the portion of Parah, because from the outset one should act in accordance with the service of the tzadik. Another reason why Parah (the service of teshuvah) follows Hachodesh (the service of the tzadik), is that the advantage of the service of teshuvah, the service from below, is due to the light that is drawn down through it. Therefore the parah adumah follows the service of the tzadik, whose service accomplishes the drawing down of the light. The above was only the first year (where Hachodesh preceded Parah). Now, however, since the light has already been drawn down, Parah can be read first.
Based on the above, parshas Parah (which often coincides with the portion of Tzav) is similar to the seven days of inauguration. We explained that the seven days of inauguration have the advantage of quantity, which represents the idea of refining the lower domain, creating many vessels. The advantage of the refining of the lower domain is the advantage of quantity. The Parsha of Hachodesh, by contrast, is connected to (and often coincides with) the portion of Shemini, because the advantage of the drawing down from above is the advantage of the light which is a qualitative advantage.
Another connection between Parah to Tzav, and Hachodesh to Shemini: During the seven days of inauguration, Moshe erected the Mishkan (which represents the elevation of the lower domain), yet they were not able to draw down the Shechinah, until the eighth day. Because, as explained above, the elevation of the lower domain can only occur after the drawing down from above. Similarly, Parshas Parah, the elevation of the lower domain, can occur only after the Hachodesh (the drawing down from above).
