Parah, Chapter 1, Mishnah 2. This masechta is devoted to the red heifer whose ashes purify a person from tumas meis, an animal that the Torah itself designates a chatas, a sin-offering. The chapter's first Mishnah clarified when a female animal has outgrown the term eglah and is properly called a parah. Our Mishnah now takes up the corresponding question on the male side: at what age is a par, a bull, fit to be offered as a korban?
The dispute
"Parim bnei shtayim" is the position of Rabbi Yosei HaGelili: a bull brought upon the mizbeiach is in its second year. His source is a pasuk in Bamidbar that speaks of taking "ufar sheini ben bakar" as a chatas. The word sheini, literally "second," he understands as pointing not to the order of the offerings but to the age of the animal: it stands in its second year.
"Vachachamim omrim: af bnei shalosh." The chachamim disagree and allow even a three-year-old bull.
Rabbi Meir widens the range further: "af bnei arba," even those in their fourth year, and "uvnei chamesh, ksheirim," five-year-olds too are fit. He adds one qualification: "ela she'ein mevi'im zekeinim mipnei hakavod." Aged animals are not actually brought, because of kavod. Leading a worn, elderly beast before Hashem is not a dignified manner of avodah, and so, however kosher such a bull may be, it is not offered.
A distinction worth noticing
That reasoning of honor belongs to the avodah of the Beis Hamikdash. The Parah Adumah, however, is not performed inside the Beis Hamikdash, so this consideration does not govern it. There, as the chapter's opening Mishnah indicated, the reason we do not let the animal grow old is entirely different: as the years pass, hairs are liable to turn black, and a black hair disqualifies the cow. In the Mikdash the concern is the honor due to the korban; with the Parah Adumah the concern is the color of the hairs.
So this Mishnah leaves us with two ages and two reasons: the years within which a bull may be brought as a korban, and the reminder that what we offer to Hashem must be offered with dignity.