  In this paper we investigate the $\mathbb{Q}$-rational points of a class of
simply connected Calabi-Yau threefolds, which were originally studied by Hosono
and Takagi in the context of mirror symmetry. These varieties are defined as a
linear section of a double quintic symmetroid; their points correspond to
rulings on quadric hypersurfaces. They come equipped with a natural $2$-torsion
Brauer class. Our main result shows that under certain conditions, this Brauer
class gives rise to a transcendental Brauer-Manin obstruction to weak
approximation. Hosono and Takagi showed that over $\mathbb{C}$ each of these
Calabi-Yau threefolds $Y$ is derived equivalent to a Reye congruence Calabi-Yau
threefold $X$. We show that these derived equivalences may also be constructed
over $\mathbb{Q}$, and we give sufficient conditions for $X$ to not satisfy
weak approximation. In the appendix, N. Addington exhibits the Brauer groups of
each class of Calabi--Yau variety over $\mathbb{C}$.
