  Violation of parity symmetry in the gravitational sector, which manifests
into unequal left and right circular polarization states of primordial
gravitational waves, represents a way to test high-energy modifications to
general relativity. In this paper we study inflation within recently proposed
chiral scalar-tensor theories of gravity, that extend Chern-Simons gravity by
including parity-violating operators containing first and second derivatives of
the non-minimally coupled scalar (inflaton) field. Given the degeneracy between
different parity-violating theories at the level of the power spectrum
statistics, we make a detailed analysis of the parity violation on primordial
tensor non-Gaussianity. We show, with an explicit computation, that no new
contributions arise in the graviton bispectra if the couplings in the new
operators are constant in a pure de Sitter phase. On the other hand, if the
coupling functions are time-dependent during inflation, the tensor bispectra
acquire non-vanishing contributions from the parity-breaking operators even in
the exact de Sitter limit, with maximal signal in the squeezed and equilateral
configurations. We also comment on the consistency relation of the three-point
function of tensor modes in this class of models and discuss prospects of
detecting parity-breaking signatures through Cosmic Microwave Background
$B$-mode bispectra.
