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

Second order estimates for transition layers and a curvature estimate for the parabolic Allen-Cahn