Let O ⊃ ℝ^d be a bounded domain of class C^1,1. In L₂(O;ⁿ), a selfadjoint matrix second order elliptic differential operator BD,_ε, 0 < ε ≤ 1, is considered with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves first and zero order terms. The operator BD,ε is positive definite; its coefficients are periodic and depend on x/ε. The behavior of the operator exponential e ^-BD,ε^t, t > 0, is studied as ε → 0. Approximations for the exponential e ^-BD,ε^t are obtained in the operator norm on L₂(O;ⁿ) and in the norm of operators acting from L₂(O;ⁿ) to the Sobolev space H¹(O;ⁿ). The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.