  The tests of parity symmetry in the gravitational interaction is an
attractive issue in the gravitational-wave astronomy. In the general theories
of gravity with parity violation, one of the fundamental results is that the
primordial gravitational waves (PGWs) produced during the slow-roll inflation
is circularly polarized. In this article, we investigate the polarization of
PGWs in the recently proposed ghost-free parity-violating gravity, which
generalizes the Chern-Simons gravity by including the higher derivatives of the
coupling scalar field. For this purpose, we first construct the approximate
analytical solution to the mode function of the PGWs during the slow-roll
inflation by using the uniform asymptotic approximation. With the approximate
solution, we calculate explicitly the power spectrum and the corresponding
circular polarization of the PGWs analytically, and find that the contributions
of the higher derivatives of the coupling scalar field to the circular
polarization is of the same order of magnitude as that of Chern-Simons gravity.
The degree of circular polarization of PGWs is suppressed by the energy scale
of parity violation in gravity, which is hardly detected on the basis solely of
two-point statistics from future cosmic microwave background data.
