Parah, Chapter 3, Mishnah 4. This chapter walks us through the elaborate preparations that preceded the burning of the parah adumah: the Kohein who was set apart for seven days, the children who were raised in guarded purity so that they could draw the water for the sprinkling, and the repeated sprinklings performed on the Kohein during those days. Our mishnah now teaches a principle that governs all of this preparation: every single preparation stands only for the specific parah for which it was made, and nothing carries over.
The mishnah opens with a pair of parallel rulings. First: "Lo hayu osin chatas al gabei chatas", the preparations carried out for one chatas were never allowed to serve a second chatas. (The parah adumah is designated a chatas.) The second clause continues in the same idiom: "velo tinok", and likewise no child, "al gabei chaveiro", was rendered fit by relying on the fitness of another boy.
What does this mean in practice? Suppose the Kohein and everything associated with him were rendered pure with a particular parah in mind, and that parah then turned out not to be usable. The purity that was achieved does not simply transfer to a different parah. Nothing may be carried over; the entire process must be begun again from the start for the new cow.
The same rule applies to the children. Picture two boys. One of them had been watched from birth specifically for the sake of the parah adumah, so that he would be fit to draw and carry the water. The second boy was watched only in a general way, and was brought along simply as a companion for his friend, without the parah in mind. The second child is not made fit by the fitness of the first. His friend's careful guarding accomplishes nothing on his behalf, because each child's preparation counts only for the purpose for which it was undertaken.
Regarding these boys the Tannaim are divided. One view holds that they themselves had to receive sprinkling from the mei chatas before they were qualified to serve, and the mishnah presents this position as "utzrichin hayu hatinokos lehazos", attributing it to "divrei Rebbi Yose Haglili". The other view is introduced with "Rebbi Akiva omer": in his opinion "lo hayu tzrichin lehazos", no sprinkling was called for. Boys shielded from birth in a way that ruled out any possibility of tumas meis carried no corpse impurity for the sprinkling to address, so haza'ah would have accomplished nothing. This second position is the accepted halacha.
One point, however, is agreed upon by both opinions: the children did require immersion in a mikveh. Even though corpse impurity was not a concern, it was entirely possible that a child had come into contact with a sheretz, a creeping creature, whose tumah is removed through immersion. So while the sprinkling was dispensed with according to Rebbi Akiva, the immersion was not.
The lesson of this mishnah is a striking one. In the service of the parah adumah, purity is never generic. It is prepared with intent, for a particular parah and for a particular child, and where that intent is absent, all the care in the world does not substitute for it.