The pairs $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}\text{ and }\overrightarrow{\mathrm{c}},\overrightarrow{\mathrm{d}}$ each determine a plane. Then the planes are parallel if 

$\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}$ is a vector perpendicular to the plane  $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ Similarly $\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{d}}$ is a vector perpendicular to the plane containing $\overrightarrow{\mathrm{c}} \mathrm{and} \overrightarrow{\mathrm{d}}$. 

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. Thus

$(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times (\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{d}}))=\overrightarrow{0}$