  We investigate quantitative estimates in homogenization of the locally
periodic parabolic operator with multiscales
  $$ \partial_t- \text{div} (A(x,t,x/\varepsilon,t/\kappa^2) \nabla ),\qquad
\varepsilon>0,\, \kappa>0. $$ Under proper assumptions, we establish the
full-scale interior and boundary Lipschitz estimates. These results are new
even for the case $\kappa=\varepsilon$, and for the periodic operators $
  \partial_t-\text{div}(A(x/\varepsilon, t/\varepsilon^{\ell}) \nabla ),$
$0<\varepsilon,\ell<\infty, $ of which the large-scale Lipschitz estimate down
to $\varepsilon+\varepsilon^{\ell/2}$ was recently established by the first
author and Shen in Arch. Ration. Mech. Anal. 236(1): 145--188 (2020). Due to
the non-self-similar structure, the full-scale estimates do not follow directly
from the large-scale estimates and the blow-up argument. As a byproduct, we
also derive the convergence rates for the corresponding initial-Dirichlet
problems, which extend the results in the aforementioned literature to more
general settings.
