= Solution
With $s=\sin\theta_0$ and $c=\cos\theta_0$, conversion from <spherical coordinates> gives $\mathbf a=r_0(s,0,c)$, $\mathbf b=r_0(-s/2,\sqrt3s/2,c)$ and $\mathbf c=r_0(-s/2,-\sqrt3s/2,c)$. The <volume> of the <parallelepiped> is the <absolute value> of their <scalar triple product>:
$$
\boxed{V=|\det(\mathbf a,\mathbf b,\mathbf c)|=\frac{3\sqrt3}{2}r_0^3\sin^2\theta_0\,|\cos\theta_0|.}
$$
The absolute value matters when the polar angle is in the lower hemisphere. At the poles or equator the three edge vectors are linearly dependent and the volume vanishes.
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