1) The source that the first Parah Adumah was burned on 2 Nisan is Rashi in Gitin (60b, DH u'Parshas Parah; see also Mitzpeh Eisan there), who writes that this is stated in the Talmud Yerushalmi 3:5. Rashi writes that the Parah Adumah could not have been burned before then, because the verse (Bamidbar 19:3) states that the blood of the Parah must be sprinkled opposite the Ohel Moed, so it follows that before the Ohel Moed (i.e. the Mishkan) had been set up (on 1 Nisan) the blood could not have been sprinkled there, and consequently the Parah could not have been slaughtered.

2) However, there was a big difference between the time of the erecting of the Mishkan and the time of the Chashmona'im. In the time of the Chashmona'im the entire Beis ha'Mikdash had been contaminated by the Greeks. In contrast, in the time of the Milu'im, the Mishkan itself was Tahor and, in additon, there were very few people who were Tamei. This can be learned from the Gemara in Sukah (end of 25a), which cites a Beraisa discussing who were the people who were Tamei at the time of the Korban Pesach in the second year (as recorded in Bamidbar 9:6). One opinion is that they were those who carried the coffin of Yosef. Another is that they were Misha'el and Eltzafan, who became Tamei when Nadav and Avihu died. Rebbi Yitzchak (on 25b) says that it cannot be either of these, because they could have become Tahor. Rashi (DH v'Im) writes that they could have become Tahor after the Parah Adumah was burned on 2 Nisan in the second year. Therefore, we would not need to resort to Tum'ah Hutrah b'Tzibur, because only a very small minority of Klal Yisrael was Tamei and all the utensils of the Mishkan were Tahor. In contrast, in the time of the Greeks the entire Beis ha'Mikdash was Tamei.

Kol Tuv,

D. Bloom