  The parabolic Allen-Cahn equation is a semilinear partial differential
equation linked to the mean curvature flow by a singular perturbation. We show
an improved convergence property of the parabolic Allen-Cahn equation to the
mean curvature flow, which is the parabolic analogue of the improved
convergence property of the elliptic Allen-Cahn to minimal surfaces by Wang-Wei
and Chodosh-Mantoulidis. More precisely, we show if the phase-transition level
sets are converging in $C^2$, then they converge in $C^{2,\theta}$. As an
application, we obtain a curvature estimate for parabolic Allen-Cahn equation,
which can be viewed as a diffused version of Brakke's and White's regularity
theorem for mean curvature flow
