Let (Formula presented.) be a bounded domain of class (Formula presented.). In (Formula presented.), we consider matrix elliptic differential operators (Formula presented.) and (Formula presented.) of order 2p ((Formula presented.)) with the Dirichlet or Neumann boundary conditions, respectively. The coefficients of (Formula presented.) and (Formula presented.) are periodic and depend on (Formula presented.), (Formula presented.). The behavior of the operator (Formula presented.), (Formula presented.), for small (Formula presented.) is studied. It is shown that, for fixed (Formula presented.), the operator (Formula presented.) converges in the (Formula presented.) -operator norm to (Formula presented.), as (Formula presented.). Here (Formula presented.) is the effective operator with constant coefficients. We obtain a sharp order estimate (Formula presented.). Also, we find approximation for (Formula presented.) in the (Formula presented.) -norm with error estimate of order (Formula presented.). The results are applied to homogenization of the solutions of initial boundary value problems for parabolic systems.