As an extension of our previous work [38], in this article, we calculate of gravitational-wave (GW) waveforms, produced by the coalescence of compact binaries, in the most general parityviolating gravities, including Chern-Simons modified gravity, ghost-free scalar-tensor gravity, symmetric teleparallel equivalence of GR theory, Hořava-Lifshitz gravity and so on. With a unified description of GW in the theories of parity-violating gravity, we study the effects of velocity and amplitude birefringence on the GW waveforms. Decomposing the GWs into the circular polarization modes, the two birefringence effects exactly correspond to the modifications in phase and amplitude of GW waveforms respectively. We find that, for each circular polarization mode, the amplitude, phase and velocity of GW can be modified by both the parity-violating terms and parity-conserving terms in gravity. Therefore, in order to test the parity symmetry in gravity, we should compare the difference between two circular polarization modes, rather than measuring an individual mode. Combining two circular modes, we obtain the GW waveforms in the Fourier domain, and obtain the deviations from those in General Relativity. The GW waveforms derived in this paper are also applicable to the theories of parity-conserving gravity, which have the modified dispersion relations (e.g. massive gravity, double special relativity theory, extra-dimensional theories, etc), or/and have the modified friction terms (e.g. nonlocal gravity, gravitational theory with time-dependent Planck mass, etc).Gravitational waves (GWs) are always produced in the circumstances with extreme conditions (e.g. the strongest gravitational field, the densest celestial bodies, the earliest stage of the universe, the highest energy scale physics, etc), and have the weak interactions with other matters during the propagation [1][2][3]. Therefore, the GWs encode the cleanest information for these extreme conditions, and provides the excellent opportunity to study the physics in these extreme circumstances. As an example of the applications, GWs can be used to test the theory of gravity [4,5], which has become an important topic in the era of GW astronomy [6][7][8][9][10][11][12]. Although Einstein's General Relativity (GR) has been considered to be the most successful theory of gravity since it was proposed, it faces the difficulties in both theoretically (e.g. singularity, quantization, etc), and observationally (e.g. dark matter, dark energy, etc). Therefore, testing GR in various circumstance is an important topic since its birth [13][14][15][16][17][18].As well known, symmetry permeates nature and is fundamental to all laws of physics. Thus, one important method for the gravity examination is to test the symmetries in gravity. In our series of works, we focus on the testing of parity symmetry in gravity. Parity symmetry implies that a directional flipping to the left and right does not change the laws of physics. It is well known that nature is parity violating. Since the first discovery...
