Parashat Shekalim – On Rosh Hodesh Adar the mitzva of giving one’s half-shekel was announced. This collection of funds paid for the communal sacrifices beginning in the month of Nisan. By announcing the obligation publicly a full month in advance, everyone would have the opportunity to prepare his half-Shekel and to donate it before Rosh Hodesh Nisan arrived.
Parashat Para and Parashat HaHodesh – Parashat Para served as a reminder to whoever was ritually defiled to purify himself with the ashes of the Para Aduma (the Red Heifer), a process which would last one week. This was in anticipation of Rosh Hodesh Nisan, when those who lived far away would begin their journey to Jerusalem to arrive in time for Pesah. Parashat Para always occurs the week before Parashat HaHodesh – which, in turn, takes place the Shabbat before Rosh Hodesh Nissan. Parashat Para reminded everyone who was tamei met (ritually impure from contact with a corpse) that Rosh Hodesh Nisan was just around the corner and that they must begin their purification immediately.
Parshat Zakhor (the one Parasha that is not connected with the Beit HaMikdash) takes place the Shabbat before Purim, because Haman (the villain of Megilat Esther), was a descendant of Amalek, whom the Torah commands us to remember, as we read in the Parashah of Ki Tetze on Shabbat Zakhor.
Rav Yitzhak Nappaha points out that these special readings can create an unusual circumstance. When Parashat Shekalim coincides with Rosh Hodesh Adar, or Parashat HaHodesh with Rosh Hodesh Nisan, three sifrei Torah are taken out to read – the first for the regular weekly Torah portion, the second for the Rosh Hodesh reading, and the third for Parashat Shekalim or HaHodesh . Similarly, when Hanukka and Rosh Hodesh fall out on Shabbat, we read from three sifrei Torah. The Ritva explains that this is not because of an inherent need to read different topics from different sifrei Torah, but rather is out of concern for the patience of the congregation, who will be forced to wait while the Torah scroll is turned back-and-forth in order to get to the correct reading.