  We give sharp $C^{2,\alpha}$ estimates for solutions of some fully nonlinear
elliptic and parabolic equations in complex geometry and almost complex
geometry, assuming a bound on the Laplacian of the solution. We also prove the
analogous results to complex Monge-Amp\`{e}re equations with conical
singularities. As an application, we obtain a local estimate for Calabi-Yau
equation in almost complex geometry. We also improve the $C^{2,\alpha}$
regularities and estimates for viscosity solutions to some uniformly elliptic
and parabolic equations. All our results are optimal regarding the H\"{o}lder
exponent.
