  We prove trivialization of the first cohomology with coefficients in smooth
vector fields, for a class of $\mathbb Z^2$ parabolic actions on $(SL(2,
\mathbb R)\times SL(2, \mathbb R))/\Gamma$, where the lattice $\Gamma$ is
irreducible and co-compact. We also obtain a splitting construction involving
first and second coboundary operators in the cohomology with coefficients in
smooth vector fields.
