  In this paper we consider a parity violating gravity model without higher
derivatives, thus ghost free. This model is constructed from the tetrad and its
derivatives and coupled to a dynamical scalar field, like axion. It can be
reduced from the Nieh-Yan term within the framework of teleparallel gravity. We
apply this model to cosmology and investigate its consequences on cosmological
perturbation theory. We find that the coupled dynamical scalar field lost its
independent dynamics at the linear order and the parity violating term itself
behaves like a viscosity. For gravitational waves, this model produces velocity
difference between left- and right-handed polarizations, but generates no
amplitude discrepancy.
