  Varieties fibered into del Pezzo surfaces form a class of possible outputs of
the minimal model program. It is known that del Pezzo fibrations of degrees $1$
and $2$ over the projective line with smooth total space satisfying the
so-called $K^2$-condition are birationally rigid: their Mori fibre space
structure is unique. This implies that they are not birational to any Fano
varieties, conic bundles or other del Pezzo fibrations. In particular, they are
irrational. The families of del Pezzo fibrations with smooth total space of
degree $2$ are rather special, as for "most" families a general del Pezzo
fibration has the simplest orbifold singularities. We prove that orbifold del
Pezzo fibrations of degree $2$ over the projective line satisfying explicit
generality conditions as well as a generalised $K^2$-condition are birationally
rigid.
