Given that $\overrightarrow c$ is parallel to the plane containing $\overrightarrow a\:and\:\overrightarrow b$
$\Rightarrow \overrightarrow c\:\:is\:\:\perp$ to $\overrightarrow a\times\overrightarrow b$
$\Rightarrow\:\overrightarrow c.(\overrightarrow a\times\overrightarrow b)=0$
$\Rightarrow\:\left|\begin {array}{ccc}2 & 3 &-1\\1 &-2 &3\\\lambda & 1 & 2\lambda -1\end {array}\right|=0$
$\Rightarrow\:\lambda=0$