  Let $\mathcal{A}$ be a line arrangement in the complex projective plane
$\mathbb{P}^2$, having the points of multiplicity $\geq 3$ situated on two
lines in $\mathcal{A}$, say $H_0$ and $H_{\infty}$. Then we show that the
non-local irreducible components of the first resonance variety
$\mathcal{R}_1(\mathcal{A})$ are 2-dimensional and correspond to parallelograms
$\mathcal{P}$ in $\mathbb{C}^2=\mathbb{P}^2 \setminus H_{\infty}$ whose sides
are in $\mathcal{A}$ and for which $H_0$ is a diagonal.
