  Let $s \in {\mathbb N}$, $T_1,T_2 \in {\mathbb R}$, $T_1<T_2$, and let
$\Omega, \omega $ be bounded domains in ${\mathbb R}^n$, $n \geq 1$ such that
$\omega \subset \Omega$ and the complement $\Omega \setminus \omega$ have no
non-empty compact components in
  $\Omega$. We investigate the problem of approximation of solutions to
parabolic Lam\'e type system from the Lebesgue class
  $L^2(\omega \times (T_1,T_2))$ in a cylinder domain
  $\omega \times (T_1,T_2) \subset {\mathbb R}^{n+1}$ by more regular solutions
in a bigger domain $\Omega \times (T_1,T_2)$. As an application of the obtained
approximation theorems we construct Carleman's formulas for recovering
solutions to these parabolic operators from the Sobolev class $H^{2s,s}(\Omega
\times (T_1,T_2))$ via values the solutions on a part of the lateral surface of
the cylinder and the corresponding them stress tensors.
