ואלו נאמרין בלשון הקודש: מקרא ביכורים, וחליצה, ... ברכת כהנים,... ופרשת המלך... (Sotah 32a)
these are recited only in the sacred tongue, Hebrew: Bikkurim and Chalitzah and the blessings and curses that were spoken on Mount Gerizim and Mount Ebal; the Priestly Benediction; and the blessing on the Torah recited by the High Priest on Yom Kippur; and the portion of the Torah read by the king at Hakhel; and the portion recited during the ritual of a heifer whose neck is broken, and the speech of a priest who is anointed for war.
While all others are not recited from the Torah scroll, or completely by heart, the king's reading of the Torah portion on Hakhel is specifically from a Kosher Torah scroll (Rambam Hagiga 3.4):
וְחַזַּן הַכְּנֶסֶת נוֹטֵל סֵפֶר תּוֹרָה וְנוֹתְנוֹ לְרֹאשׁ הַכְּנֶסֶת וְרֹאשׁ הַכְּנֶסֶת נוֹתְנוֹ לַסְּגָן וּסְגָן לְכֹהֵן גָּדוֹל וְכֹהֵן גָּדוֹל לַמֶּלֶךְ כְּדֵי לְהַדְּרוֹ בְּרֹב בְּנֵי אָדָם. וְהַמֶּלֶךְ מְקַבְּלוֹ כְּשֶׁהוּא עוֹמֵד וְאִם רָצָה יֵשֵׁב וּפוֹתֵחַ וְרוֹאֶה וּמְבָרֵךְ כְּדֶרֶךְ שֶׁמְּבָרֵךְ כָּל קוֹרֵא בַּתּוֹרָה בְּבֵית הַכְּנֶסֶת.
The ḥazzan of the synagogue would take a Sefer Torah and hand it to the head of the synagogue, and the head of the synagogue would hand it to the deputy high priest, and the deputy high priest to the high priest, and the high priest to the king, to honor him by the service of many persons. The king would receive it standing or sitting, as he pleased. He would open it and look in it, reciting the blessing used by anyone who reads the Torah in the synagogue...
I suppose reading the Parasha means reading off the scroll word by word, not a translation or interpretation, and this is surely in Hebrew. Rambam further says that the reading makes Yotzeh even those unfamiliar with Hebrew (similar to Meggilah).
Why the Mishnah specifies that the reading should be only in Hebrew?