  This paper considers a family of second-order parabolic equations in
divergence form with rapidly oscillating and time-dependent periodic
coefficients and an interface between two periodic structures. Following a
framework initiated by Blanc, Le Bris and Lions and a generalized two-scale
expansion in divergence form of elliptic homogenization with an interface by
Josien, we can determine the effective (or homogenized) equation with the
coefficient matrix being piecewise constant and discontinuous across the
interface. Moreover, we obtain the $O(\varepsilon)$ convergence rates in
$L^{2(d+2)/{d}}_{x,t}$ with $\varepsilon$-smoothing method and the uniform
interior Lipschitz estimates via compactness argument.
