The Mishna refers to the Hakhel reading as the Parsha of the king. In his commentary on the Gemara, Rashi writes that the requirement for the king to read the Torah is derived in the Sifri on Parshas Shoftim.
Parshas Shoftim contains the Mitzvah of appointing a king and the Mitzvos that applied to him, including the Mitzvah to write a Sefer Torah. The Posuk says “he shall write this Mishneh Hatorah”. Mishneh Torah refers to the Book of Devarim, but the king had to have a complete Torah scroll. So why is it called Mishneh Hatorah?
In the second answer, the Sifri explains that it is because all of the Parshios read at Hakhel came from the Book of Devarim - Mishneh Torah.
We derive the requirement for the king to read at Hakhel from the Torah alluding to the reading of Hakhel in the Mitzvah of the king to write a Sefer Torah. According to some, the king would read from the Sefer Torah that he was instructed to write.
This would explain why the Parsha of the king in Parshas Shoftim was selected as one of the Parshios to be read at Hakhel.
