A chukat is a particular type of commandment that defies rational explanation. The parashas refers to the chukat (decree) of the parumah adumah (red heifer). The entire premise of the chukat of the red heifer points towards having emunah (faith) in the effectiveness of this remedy for contamination. In other words, it is only through a chukat that is not objectively clear to the intellect, through which a means of purification occurs. It is as if the goal of the chukat is to inspire our faith in the One who gave the decree.
The parumah adumah (red heifer) offering is slaughtered outside of the camp; it is completely burned in fire. Hyssop, cedar wood, and crimson thread are thrown into the fire with the red heifer. The ashes are used in a specific manner – only sparingly mixed with mayim chayim (living water; i.e. from a water source like a river). The purpose of this water with the mixture of ashes is to purify people who have come into contact with a dead body, and, therefore tamei (unclean). The water is sprinkled upon them on the third and the seventh day of their purification.
Paradoxically, the kohein who is tahor (clean) becomes tamei (unclean) when he performs the offering of the parumah adumah. This is the paradox: the very ashes of the red cow that cause an unclean person to become clean, cause the pure person, who prepares the ashes, to become impure. “Who can bring a clean thing out of an unclean?” (Job 14:4, JPS 1917 Tanach). Only G-d.
Additionally, the account of the passing of Miriam occurs right after the description of the chukat of the parumah adumah. The Sages infer that this exemplifies, how like an offering brings atonement, so does the death of a tzaddik (righteous person). Moreover, the passing of Aaron occurs in juxtaposition to a description of the garments of the Kohein Gadol. As the garments of the Kohein Gadol atone for sin, so does the death of a righteous person (Moed Katan 29a). The deaths of Aaron and Miriam brought atonement to the generation in the desert.