  High-order spatial derivatives are of crucial importance for constructing the
low energy effective action of a Lorentz or parity violating theory of quantum
gravity. One example is the Ho\v{r}ava-Lifshitz gravity, in one has to consider
at least the sixth-order spatial derivatives in the gravitational action, in
order to make the theory power-counting renormalizable. In this paper, we
consider the Lorentz and parity violating effects on the propagation of GWs due
to the fifth and sixth-order spatial derivatives respectively. For this purpose
we calculate the corresponding Lorentz and parity violating waveforms of GWs
produced by the coalescence of compact binaries. By using these modified
waveforms, we perform the full Bayesian inference with the help of the open
source software \texttt{Bilby} on the selected GW events of binary black hole
(BBH) and binary neutron stars (BNS) merges in the LIGO-Virgo catalogs GWTC-1
and GWTC-2. Overall we do not find any significant evidence of Lorentz and
parity violation due to the fifth and sixth-order spatial derivatives and thus
place lower bounds on the energy scales $M_{\rm LV} > 2.4 \times 10^{-16} \;
{\rm GeV}$ for Lorentz violation and $M_{\rm PV} > 1.0 \times 10^{-14} \; {\rm
GeV}$ for parity violation at 90\% confidence level. Both constraints represent
the first constraints on the fifth- and sixth-order spatial derivative terms
respectively in the framework of spatial covariant gravity by using the
observational data of GWs.

General Relativity and Quantum Cosmology;High Energy Astrophysical Phenomena;High Energy Physics - Theory