  We give a sufficient condition for birational superrigidity of del Pezzo
fibrations of degree $1$ with only $\frac{1}{2} (1,1,1)$ singular points,
generalizing the so called $K^2$-condition. As an application, we also prove
that a del Pezzo fibrations of degree $1$ with only $\frac{1}{2} (1,1,1)$
singular points embedded in a toric $\mathbb{P} (1,1,2,3)$-bundle over
$\mathbb{P}^1$ is birationally superrigid if and only if it satisfies the
$K$-condition.
