Since $\vec{a}$ and $\vec{b}$ are coplanar, therefore $\vec{a} \times \vec{b}$ is a vector perpendicular to the plane containing $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$. Similarly $\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}}$ is a vector perpendicular to the plane containing $\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{d}}$. Thus, the two planes will be parallel if their normals i.e. $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}}$ are parallel. $\Rightarrow(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}})=0$