Matters not revealed to Moshe Rabbeinu were revealed to R. Akiva...
The Parah Aduma is a paradigmatic Chukkah, a law with reasons are so obscure that not King Shelomo, not even Moshe Rabbeinu fully understood it. It is surprising then that certain Tannaim are said to have understood it better than Moshe.
Things that were not revealed to Moshe Rabbeinu were revealed to R. Akiva and his colleagues...
When Moshe went up to Above, he heard the voice of the Holy One Blessed be He (HKBH) sitting and learning Parah Adumah and repeating Halacha in the name of he who said it:- "R. Eliezer said 'egla (arufah) is 3 years-old, parah adumah is 2 years-old (Parah 1:1)." Moshe said before HKBH - may it be Your will that he may be from my descendents. He answered Moshe: "By your life he is from your lions, as it says "the name of the one is Eliezer" (Bamidbar Rabbah 19, 4).
R. Levi said: Every teaching that came out of Hashem's mouth, when he told it to Moshe, He told him its status - tahor or tamei. When He reached the section Emor (about impurities of kohanim), Moshe asked: Master of the World, if he( kohen) becomes impure, what is his purification. Hashem did not respond. At that moment, Moshe's face turned red from embarrassment.... When he reached this section of the red heifer, the Holy One Blessed be He said to Moshe... (at that time I) did not respond. (Now I answer -) this is his purification...
The Chidushei Basra on Midrash explains this passage b'derech hamusar as signifying that the difficult laws of parah adumah cannot be assimilated in ease and comfort. Moshe had to experience embarrassment and shame in order to be able to understand the matter of this purification. He proceeds to explain that, of all tannaim, R. Akiva is one who experienced the greatest embarrassment to secure his learning. He could, therefore, understand more about this topic than Moshe Rabbeinu.
This interpretation is on to something, although I would suggest that it be broadened to consider the most salient feature of parah adumah as a chok - what makes it so much of a paradigm. There are, after all, many other laws that we do not understand; yet this one is the one that classically represents chukkim in the literature of Chazal and in popular consciousness.
Going back to the Midrash Rabbah 19:1, we find a specific feature of this law that is emphasized as what makes it a chukkah.
This is what it states: Who will bring pure from impure, not one (Iyov 14). Like Avraham who came from Terach, Chizkiah from Achaz, Yoshiah from Amon... It was taught: those who are occupied with the red heifer from beginning to end make their clothes unclean. It itself (contrarily) purifies defiled garments. HKBH said: I set a rule, I decreed a decree, you cannot transgress my decree.
The mystery of how the same agent can serve to purify and make unclean, redeem and condemn, elevate and debase is at the core of Parah Adumah. Who had not experienced a tragedy or a disappointment that ultimately lead to profound spiritual growth? The contrary is also true. How often is an apparent stroke of good luck a cause of ultimate loss?
This message of Parah Adumah is personified by two Tannaim.
R. Eliezer is, of course, famously associated with another heifer. As told in Breshis Rabbah Ch. 42, 1, as a young man of 22, R. Eliezer had not yet studied Torah. The cow with which he was plowing fell down and broke its leg. "He said: for my good did my cow broke its leg". Motivated, R. Eliezer run away to study Torah with R. Yochanan ben Zakkai. R. Eliezer's misfortune was the cause of his elevation.
But how is this relevant to R. Akiva?
The phrase "R. Akiva and his colleagues" echoes that of Sanhedrin 100b - "R. Akiva and his colleagues who have handed themselves over to slaughter for words of Torah". Their heroism, their unshakeable conviction that something good will sprout from their martyrdom is the real meaning of the laws of parah adumah. R. Akiva and his colleagues who sacrificed themselves in the unshakeable belief that pure does come out of impure, that bad things that happen to good people are the genesis of much greater good and that Torah is gained through suffering, were revealed the true pshat in parah aduma.
1 See Bamidbar Rabbah 19,4
2 In the original context in Shemos 18, one of two sons of Moshe.
3 As a fourty year old beginning his Torah study in a classroom with little boys, for example. The difficulty with this suggestion is that it does not account for inclusion of R. Akiva's colleagues in the source as we find it in the Midrash.
4 See Sippurno ibid.
5 The version told in the beginning of Pirkei DR. Eliezer does not describe a cow.