Q. Does an avel sitting shiva read the parsha shnaim mikra, since he is prohibited in Torah learning?

A. Kaf Hachaim (285: 27), Yalkut Yosef (7: p. 136) permit as it is an obligation akin to reading Shema that is allowed.
Maharshal, P’ri Megodim (Eishel Avrohom 285: 6) and others permit on the Shabbos of the shiva itself. While Kerem Shlomo permits from Erev Shabbos after chatzos.
Horav Shlomo Miller’s opinion is it is better to wait until Shabbos.
Some Poskim add that although one may usually also include Rashi, on this Shabbos of shiva he should abstain. (Orchois Yosher Y.D. 1: 28 ”“ Habais Hayehudi 3: 36: 15). The Rov’s opinion is similar, but it should be learned after the shiva.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a