The third of the Arba Parshiyot -- the four special sections read as the Maftir in the weeks surrounding Purim and Pesach, Parshat Parah (Chukat Chapter 19), read this week, concerns the laws of the Red Heifer, the ashes of which were used to purify one who came into contact with the dead. Rashi (Megilla 29a) explains that the reason that this portion is read before the month of Nissan is that all of Klal Yisrael had to purify themselves in preparation for the Korban Pesach -- the Paschal Sacrifice. Even though, to our great sorrow, we are currently not able to bring korbanot, we anxiously look forward to the return of the sacrificial order and therefore remind ourselves of the laws connected to it.
Several Rishonim (earlier authorities) maintain that just as the reading of Parshat Zachor is min haTorah -- biblically mandated -- so too is the reading of Parshat Parah (see Ritva (Megilla 17b), Tosfot HaRosh (B'rachot 13a)). Many Acharonim (later authorities) question the biblical source for such a command (see Magen Avraham (O.C. 685), Bai'ur haGra (ibid)); several offer various suggestions (see Meshech Chachma (Chukat s.v. "hinei"), Birchat Peretz (Chukat)). I have heard from Rav Hershel Schachter shlit"a that the source might be from the mitzva to remember the Cheit Ha'eigel -- the sin of the Golden Calf. In Parshat Eikev (9:7) we read: "Z'chor al tishkach eit asher hiktsafta et Hashem Elokecha b'Choreiv" -- "Remember, do not forget, that which you angered the Lord, your G-d at Choreiv (Sinai)," a reference to the Golden Calf. Rashi (Chukat 19:22) quotes from R. Moshe HaDarshan that the Red Heifer served as an atonement for the sin of the Golden Calf. He uses an allegory concerning a child who dirtied the king's palace. "Let the mother (the Heifer) clean up the mess made by her son (the Golden Calf)." Rashi mentions various aspects of the processing of the Parah Aduma that highlight this connection. By reading Parshat Parah, we thus fulfill our obligation to remember the Cheit Ha'eigel.
On a halakhic (Jewish legalistic) level, perhaps we can suggest that the source that the Cheit Ha'eigel must be mentioned orally and not just remembered in our minds is derived through halakhic exegesis similar to that which is applied to the mitzva of remembering Amalek, fulfilled through the reading of Parshat Zachor. The Sifrei (D'varim 25:17) comments on the passage "Zachor eit asher asah l'cha 'Amalek ... lo tishkach!" -- "Remember that which Amalek did to you... do not forget!" as follows: "I might think (you should remember) in your heart, but "do not forget" indicates memory in the heart; what does "Remember" come to teach you? -- with the mouth (to mention it verbally)." According to the aforementioned analysis of the source for the reading of Parshat Parah, the Torah indicates an obligation to orally remember the sin of the Golden Calf. Concerning the Cheit Ha'eigel as well, the Torah states "1) Remember, 2) do not forget." This would seem to imply an obligation to state it verbally and not just remember it mentally.
On a hashkafic (philosophical/ethical) level, I have heard an explanation* as to how the Parah Aduma serves as an atonement for the Cheit Ha'eigel. According to many commentaries (see Ramban, Ibn Ezra, Malbim), the sin of many of the Jewish people in the episode of the Golden Calf was not that they worshiped the Calf as a deity but rather that they used it as a point of focus to worship G-d. Indeed, the keruvim, the cherubs on the aron kodesh (holy Ark), would serve a similar purpose. The sin lay in the fact that G-d had not permitted the image of a calf to be used to achieve this result, and therefore, such service bordered on idolatry. The antidote to the sin of serving G-d in a fashion which he had not commanded would be a wholehearted commitment to chukim, those laws which were commanded but do not have revealed reasons. By faithfully observing the chukim, Jews demonstrate that mitzvot, expressions of the Divine Will, must be observed regardless of our ability to understand them, and that we cannot invent other forms of Divine service even if, according to our own understanding of the mitzvot, we think that they can achieve the same result.
This approach still leaves us with a question. If the source for reading Parshat Parah is ultimately remembrance of Cheit Ha'eigel, why not read Parshat Ki Tissa, which directly describes the sin itself? Perhaps we can suggest an answer based on a thought expressed by Rav Wolbe shlit"a. When making an assessment of one's weaknesses in preparation for t'shuva (repentance), it is not sufficient to list all the chat'aim (sins) that one has committed. If one only focuses on his deficiencies and moral lapses, how can he be expected to change? What tools can he harness to better himself? Therefore, a person must also make a careful accounting of his positive qualities, those which he can marshal to overcome his deficiencies. It would appear that in the commandment to remember the sin of the Golden Calf, G-d directs us to not only remember our past disastrous error, but to focus on methods with which we can overcome and atone for this great sin. By reading Parshat Parah, symbolizing the full, unquestioning commitment to the d'var Hashem --- the word of G-d -- we stress not the problem but the solution. As we just read in Megilat Ester (9:27), as elaborated upon by Chazal (Shabbat 88a) "Kiy'mu v'kib'lu hay'hudim" -- the Jewish people, in the days of Mordechai and Esther, reaffirmed their commitment to the entirety of Torat Hashem. May we merit to always maintain this state of devotion
*I would welcome being reminded of the source of this idea.