  This paper is concerned with quantitative homogenization of second-order
parabolic systems with periodic coefficients varying rapidly in space and time,
in different scales. We obtain large-scale interior and boundary Lipschitz
estimates as well as interior $C^{1, \alpha}$ and $C^{2, \alpha}$ estimates by
utilizing higher-order correctors. We also investigate the problem of
convergence rates for initial-boundary value problems.
