Why read parshas parah davka on Shabbos? Why not during the week just before rosh chodesh Nisan? (The same question is asked with respect to all of the 4 parshiyos, as they could all just as well have been done during the week.)
Both in last week's parsha and in this week's parsha we learn the halacha that building the mishkan is not doche Shabbos. Rashi in last week's parsha explains that the word "ach" in "ach es shabsosai tishmoru" is a miyut that excludes work on the mishkan from being done on Shabbos. Chazal explain that the juxtaposition of Shabbos and mishkan in this week's parsha teaches the same idea. (Why you need two limudim is not my topic -- that's a question for homework : )
Sefas Emes on last week's parsha explains: cheit ha'eigel tainted all of creation. The world post-cheit ha'eigel was a different world; it was like the world of Adam after the cheit, after gan eden was no more. However, the cheit could not taint Shabbos -- Shabbos stands apart from the other six days of creation and is the nekudah pnimit that can never be sullied. Building the mishkan was a tikun for the cheit ha'eigel. On Shabbos, you don't need that tikun -- you don't need to do meleches hamishkan to effect your tikun because on Shabbos you are in a state that needs no tikun, that is unaffected by sin. Therefore, meleches hamisihkan is prohibited on Shabbos.
The Midrash tells us that the secret of parah adumah was understood only by Moshe. We can't relate to it. Sefas Emes explains that Moshe alone was absent from the camp during cheit ha'eigel and had nothing to do with it. Moshe was untainted by sin; therefore, only he stood on the level necessary to learn parshas parah.
On Shabbos we too return to that untainted level. The crowns we lost due to cheit ha'eigel are returned to us on Shabbos -- we are as if we are in the pre-cheit stage. Therefore, it is davka on Shabbos that we read parah, as davka on Shabbos we have the ability to understand a little more deeply, a little like Moshe, what the parsha is all about.