  In this paper we study the quantitative homogenization of second-order
parabolic systems with locally periodic (in both space and time) coefficients.
The $O(\varepsilon)$ scale-invariant error estimate in $L^2(0, T;
L^{\frac{2d}{d-1}}(\Omega))$ is established in $C^{1, 1}$ cylinders under
minimum smoothness conditions on the coefficients. This process relies on
critical estimates of smoothing operators. We also develop a new construction
of flux correctors in the parabolic manner and a sharp estimate for temporal
boundary layers.
