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We deal with a class of semilinear parabolic PDEs on the space of continuousfunctions that arise, for example, as Kolmogorov equations associated to theinfinite-dimensional lifting of path-dependent SDEs. We investigate existenceof smooth solutions through their representation via forward-backwardstochastic systems, for which we provide the necessary regularity theory.Because of the lack of smoothing properties of the parabolic operators at hand,solutions in general will only share the same regularity as the coefficients ofthe equation. To conclude we exhibit an application to Hamilton-Jacobi-Bellmanequations associated to suitable optimal control problems.

Semilinear Kolmogorov equations on the space of continuous functions via BSDEs