© 2015, Springer Science+Business Media New York. Let [InlineMediaObject not available: see fulltext.] be a bounded C1,1 domain. In [InlineMediaObject not available: see fulltext.] we consider strongly elliptic operators AD,ɛ and AN,ɛ given by the differential expression b(D)*g(x/ɛ)b(D), ɛ > 0, with Dirichlet and Neumann boundary conditions, respectively. Here g(x) is a bounded positive definite matrix-valued function assumed to be periodic with respect to some lattice and b(D) is a first-order differential operator. We find approximations of the operators exp(−AD,ɛt) and exp(−AN,ɛt) for fixed t > 0 and small ɛ in the L2 → L2 and L2 → H1 operator norms with error estimates depending on ɛ and t. The results are applied to homogenize the solutions of initial boundary value problems for parabolic systems.