  We deal with a class of semilinear parabolic PDEs on the space of continuous
functions that arise, for example, as Kolmogorov equations associated to the
infinite-dimensional lifting of path-dependent SDEs. We investigate existence
of smooth solutions through their representation via forward-backward
stochastic 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 of
the equation. To conclude we exhibit an application to Hamilton-Jacobi-Bellman
equations associated to suitable optimal control problems.
