Let O ⊂ Rd be a bounded domain of class C1,1. Let 0 < ε - 1. In L2(O;Cn) we consider a positive definite strongly elliptic second-order operator BD,ε with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (BD,ε − ζQ0(·/ε))−1 as ε → 0. Here the matrix-valued function Q0 is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L2(O;Cn)-operator norm and in the norm of operators acting from L2(O;Cn) to the Sobolev space H1(O;Cn) with two-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q0(x/ε)∂tvε(x, t) = −(BD,εvε)(x, t).