The following is a brief summary of some of the thoughts said over by R' Frand on the parsha this evening. I have attempted to reproduce these vorts to the best of my ability. Any perceived inconsistency is the result of my efforts to transcribe the shiur and should not be attributed to R' Frand.
Rashi quotes the Medrash Tanchuma who compares the Parah Adumah with the following parable. A child, the son of the king's cleaning lady, had soiled the palace and the mother came and cleaned up after her child. Rashi explains that the Parah Adumah is coming to be m'chaper and atone for the Cheit HaEgel.
But the obvious questions is, other than the two being cows, what does the Parah Adumah have to do with the Egel?
R' Frand first quoted the Klei Yakar who explains that when the Jews received the Torah at Har Sinai they were on a high spiritual level and had been cleansed of their tumah. However, when they sinned with the Egel, the tumah returned.
R' Frand clarified that this was not to say that if the Jews had not sinned there would not have been any death. People still would have died, but their death would not have been at the hands of the Malach HaMaves, they would have died by Neshikah. But because they sinned with the Egel, tumah came back through the death at the hands of the Malach HaMaves. So the Parah Adumah functions to be m'chaper for the Egel, because it cleanses us from the tumah.
R' Frand quoted another answer from the Meorei Ohr. He first quoted Rashi who states that prior to the Egel the Jews were called Temimim. But as a result of the Egel, they had blemishes and needed to be purified.
The Meorei Ohr explained that Rashi's use of the word Tamim means that a person should just accept what Hashem does. But when the Jews made the Egel they were trying to outsmart Hashem. They were desperate without Moshe, so they took things into their own hands and made the Egel so that Hashem could speak with them through the Egel.
But the Jews should have accepted what Hashem had set as the path and been Tamim in this regard. Instead they tried to figure a work around.
The Parah Adumah is the antithesis or perhaps the antidote for the Egel. The Parah Adumah is beyond our understanding as it is a Chok. By following the rules of Parah Adumah, we say to Hashem ,we don't get it and we are not going to try figure it out. Instead, we are putting our trust in You that this the path that we should follow.
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