I think it is a reasonable observation that Purity does not have the resonance for modern Jews as, to use other “P” words, Peoplehood or Prophecy. We are comfortable with self-definitions that emphasize national structure or revealed ethical norms, but describing Judaism as the struggle to separate oneself from impure forces sounds, well, a little too religious to be Jewish. And while most of the Torah, beginning midway through Exodus and running almost all the way through Numbers is a narrative of the struggle that flows from the requirement of purity, these passages in our times have nothing close to the cache of those from Genesis, Deuteronomy, or the first half of the book of Exodus.
The tradition seems aware of our preference for other matters and goes out of its way to emphasize the Torah’s teachings of purity. The very first page of the Talmud is a consideration of the temporal parameters for reciting the Shema, the Shema being one of those prophetic/national statements that we cherish and with which we are comfortable. Like any question of timing in antiquity, an astronomical answer would suffice – we know it is evening when a certain number of stars are out or when darkness attains a given level. But the Talmud goes out of its way to teach the timing of the evening Shema in terms of the purification ritual of the Kohanim (which itself has a diurnal component) in order that considerations of purity be taught alongside those of revelation and peoplehood. In the Talmud, and as is the case of the Exodus narrative of the aseret ha-dibrot, purity precedes prophecy.
This week our journey toward redemption takes up the importance of purification as we read the third of the four special readings which precede Pesach. The order of these parshiyot is given in Mishnah Megilla 3:4:
ראש חדש אדר שחל להיות בשבת קורין בפרשת שקלים חל להיות בתוך השבת מקדימין לשעבר ומפסיקין לשבת אחרת בשניה זכור בשלישית פרה אדומה ברביעית החדש הזה לכם בחמישית חוזרין לכסדרן לכל מפסיקין בראשי חדשים בחנוכה ובפורים בתעניות ובמעמדות וביום הכפורים
On Rosh Hodesh Adar which falls on Shabbat we read the section Shekalim. If it falls in the middle of the week we advance [Shekalim] to the previous Shabbat and skip ahead to the next Shabbat when we read Zakhor for the second [of the four special Shabbatot]. We read Parah Adumah for the third and HaHodesh Hazeh Lakhem for the fourth. 
This week’s reading, Parashat Parah (Numbers 19), describes the fulcrum of our ancient purification rituals, which is the production of cleansing water formed from the addition of ashes from an unblemished red cow which is sacrificed and burnt “outside the camp”. The water is mixed with the ashes and sprinkled (by means of hyssop) on those who have contracted the impurity which arises from the contact or encounter with a dead body. This section is read in preparation for Pesach so that those who have contracted such impurity can participate in the Pesach ritual in a pure state.
In the journey to Pesach we note some obvious calendric parallels between Judaism and Christianity. The nexus between Passover and Easter is well-known. The connections between Purim (which we introduce with Zakhor) and Carnival/Mardi Gras stand out at least enough to make us uncomfortable, and to make “Purim is not Mardi Gras” a robust sermonic maneuver. But is Shabbat Parah our Ash Wednesday?
There are at least some superficial similarities. There is the trajectory (Purim/Parah/Pesach versus Carnival/Ash Wednesday/Easter). Shabbat Parah is typically read on the Shabbat immediately following Purim – it is postponed this year because Rosh Hodesh Nisan falls on Shabbat, but usually the admonishment for purity follows on the heels of the purge. There are as well the ashes of each tradition, which besides being outward signs of mourning and repentance are momento mori – reminders of mortality which is the primal impurity to which the ritual of the red cow comes to address. As we are along our respective journeys to national redemption or personal salvation, we are called to confront our individual natures as mortal beings. And there is also the parallel of nullification – just as the Lenten season requires abstention and moderation so Jews begin the cleaning and putting aside of leaven which will culminate in the bitul hametz on the eve of Pesach. For one who wanted to make a connection between Ash Wednesday and this Shabbat’s Torah reading, there would seem on what to stand.
An interesting liturgical point of contact is the use of Psalm 51  as a reading in the Christian Ash Wednesday service and as a suggested recitation for Jews on Shabbat Parah (at least by the Artscroll Tehillim). It opens with the David being confronted by Natan on his sin with Batsheva and the relevant verses are:
Psalm 51:4-9 4 Wash me thoroughly of my iniquity, and purify me of my sin; 5 for I recognize my transgressions, and am ever conscious of my sin. 6 Against You alone have I sinned, and done what is evil in Your sight; so You are just in Your sentence, and right in Your judgment. 7 Indeed I was born with iniquity; with sin my mother conceived me. 8 Indeed You desire truth about that which is hidden; teach me wisdom about secret things. 9 Purge me with hyssop till I am pure; wash me till I am whiter than snow. (NJPS)
This point of contact between the traditions raises the very problematic issue of the relationship between impurity and sin, and the differing resolutions to that tension which are found across and between Christianity and Judaism . But at the very least our short exploration should make us appreciate Shabbat Parah’s important role among the four special Shabbatot which lead us to Pesach. Even if we naturally groove to the exhilaration of escape which we experience on Purim and to the anticipation of redemption, it is well that we are reminded that we go through our liturgical program as flesh and blood individuals who were born, can give birth, will experience sickness and will die. And that confrontation of physical beings who are subject to generation and corruption with a God who is eternal requires a sophisticated theological response, which is happily one to which the center and bulk of Torah is devoted.
 The frequent uses of [p] are for alliterative purposes only, and should not be taken as an endorsement of Wellhausen-style higher criticism. (Though I do endorse it).
 The Mishnah goes on to say that on the fifth Shabbat “we return to the order” i.e., we either revert to the regular practice of reading the maftir from the parasha of the week or we resume the reading of the parasha of the week which was disrupted by the reading of the special parshiyot. This is a controversy in the Talmud (TB Megilla 30b) and its implications for our understanding of the historical development of the public Torah reading is explored in R. Sperber’s Minhagei Yisrael, Vol. 1 p. 88.
 It is Psalm 50 in the Christian order.
 A comprehensive overview can be found in my friend Jonathan Klawans’ Purity and Sin in Ancient Judaism