Let the pair of vectors $\vec{a}, \vec{b} $ and $\vec{c}, \vec{d}$ each determine a plane. Then, the planes are parallel, if

$(\vec{a}×\vec{b})×(\vec{c}×\vec{d})=\vec{0}$

Let $\vec{n_1}$ and $\vec{n_2}$ be the vectors normal to the planes determined the pairs of vectors $\vec{a}, \vec{b} $ and $\vec{c}, \vec{d}$ respectively.

Then,

$\vec{n_1}= \vec{a}×\vec{b} $ and $\vec{n_2} = \vec{c}×\vec{d}$

If the planes are parallel, then their normals are parallel.

$∴\vec{n_1}×\vec{n_2}= \vec{0}⇒(\vec{a}×\vec{b})×(\vec{c}×\vec{d})=\vec{0}$