Parallel lines: $2x+6y+c=0$. Intercept points: $( -\tfrac{c}{2},0)$ and $(0,-\tfrac{c}{6})$. Distance between them $=\sqrt{\left(\tfrac{c}{2}\right)^{2}+\left(\tfrac{c}{6}\right)^{2}} =|c|\,\dfrac{\sqrt{10}}{6}$. Set equal to $10$ ⇒ $|c|=6\sqrt{10}$ ⇒ two values $c=\pm6\sqrt{10}$.