We don't have the parah adumah anymore, but there are ways to attain purity even in our generation.
The ashes of the parah adumah were sprinkled on the tamei person on the third and seventh day after he became tamei.
The Be'er Mayim Chaim zt'l explains that the third day represents Torah, and the seventh day symbolizes Shabbos. Because learning Torah and keeping Shabbos purifies the Jewish nation from the tumah caused by aveiros.
We begin with the purity of the Torah, and then we will discuss the purity of Shabbos.
Purity of Torah
The Shlah (Chukas) writes, "Although we don't have the ashes of the parah adumah anymore, a person can purify himself when he attaches himself to studying Torah, teaching it, and keeping it."
We say in Akdamos, צבי וחמיד ורגיג דילאון בלעותא, "Hashem loves and desires that we should become toil over Torah study."
The Gemara (Brachos 63:) says, "How do we know that Torah knowledge exists only among those who are moser nefesh to know Torah? Because it states (this week's parashah, 19:14) זאת התורה אדם כי ימות באהל, "Where do you find Torah? By those who die in the tents of Torah."
To "die in the tents of Torah" means to study Torah with all your strength.
When a chasan says a drashah, it is the custom in many communities that people sing in to interrupt and prevent him from completing his drashah. The Pnei Menachem zt'l said that this custom teaches the chasan, "Until now, you were single, and you didn't have disturbances. When you wanted to study Torah, nothing could stop you from doing so. But now you are married, and you have obligations. Sometimes, you will begin studying Torah, and suddenly you will be interrupted. You need to get accustomed to studying Torah, despite all the interruptions."
A hint is from the pasuk, אדם כי ימות באהל, that even when you are in an אהל, in your home where you have disturbances and interruptions, even then strive to be, זאת חקת התורה אדם כי ימות, to sacrifice yourself for the goal of learning and knowing Torah.
Chazal (Shabbos 31) say that we will be asked in heaven, קבעת עיתים לתורה, "Did you set aside times for Torah?" One translation of קבע is to steal (see Mishlei 22:23). The Kotzker zt'l said that we will be asked, קבעת עיתים לתורה, did you steal time away from your busy schedule to study Torah?
Rebbe Aharon Roth zt'l (Shomer Emunim) writes in a letter that there was a time when he tasted death in every word of Torah and tefillah. He was going through a spiritually difficult time, and he couldn't say a single word without feeling a most bitter experience. One day, he went to a tzaddik's grave, and prayed there with tears that he should pass this difficult test. Then, he took out a Gemara and opened it at a random place. It opened to the Rosh in hilchos mikvaos. He told himself, "I'm not leaving this spot until I understand this." It was bitter, but he kept on studying. And then he tasted the sweetness of Torah, and that difficult test left him.
This is hinted in the words, זאת התורה, the acquisition of Torah is, אדם כי ימות באהל, when you allow yourself to feel the bitter taste of death, and even then, you don't stop learning. You keep on studying until the Torah becomes sweet.
Torah study requires toil, but that is for our benefit. As it states, לפום צערא אגרא, the reward is in accordance with the hardship.
It states (Avos 5:5) עומדים צפופים ומשתחוים רווחים. The Ra'v Bartanura zt'l explains, "צפופים comes from the word צף על המים, to float. So many people pressed together in the Beis HaMikdash, until their feet were raised above the ground, and they stood in the air. Nevertheless, משתחוים רווחים, when they bowed down, a miracle occurred; they bowed with plenty of space, and each person was four amos away from his fellow man. This way, no one would hear his friend's viduy, when he repents on his aveiros."
The Chasam Sofer zt'l asks that Hashem could have made a miracle that there should be enough place for everyone even when they stood upright. What was the benefit of standing tightly together in the Beis HaMikdash?
The answer is that this increased their reward. As Chazal (Brachos 6:) say, אגרא דכלה דוחקא, the reward for attending a crowded Torah shiur is for being squeezed by the many people present.
Similarly, for all those who have a hard time learning Torah, we tell them that this is for their benefit. Because of their hardships, they will earn a greater reward.
The Magen Avraham (end of 580) writes, "Friday parashas Chukas, some people fast because on this day, twenty wagons filled with sefarim were burned in France. The fast wasn't established on the date of the month that this tragedy occurred; instead, the fast is on the Friday erev Shabbos Chukas, because a shaalas chalom (a prophetic dream) revealed that the tragedy was associated with the parashah. Because on the words זאת חקת התורה, Onkelus writes, דא גזירת אורייתא, this is the decree on the Torah.
"Another reason for this fast is that on this day, two large communities were destroyed in the massacres of the years ת"ח ת"ט, as written in the selichos authored by the Shach."
Today, most people don't fast on this day. Nevertheless, since the Torah was burned, r'l, on this day, we should utilize this day to accept on our shoulders the yoke of the Torah, to love Torah, and to make its study our life's mission. The degree of our success isn't as important as our efforts to learn and know as much as possible.
The Gemara (Pesachim 42:) mentions עמילן של טבחים, which is dough made from not fully grown wheat kernels. This dough was placed on top of a cooking pot to draw out the זוהמא, the impure particles found in the food.
Tzaddikim said that a more common translation of עמילן is toil. Thus, Chazal are hinting that toiling in Torah draws out and removes a person's זוהמא, and impurities.