  We compute the first differential cohomology of the orthosymplectic Lie
superalgebra $\mathfrak{osp}(2|2)$ with coefficients in the superspace of
linear differential operators acting on the space of weighted densities on the
(1,\,2)-dimensional real superspace. We also compute the same, but
$\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles
spanning these cohomology. This work is a simplest generalization of a result
by Basdouri and Ben Ammar [Cohomology of $\frak {osp}(1|2)$ with coefficients
in $\frak{D}_{\lambda,\mu}$.
