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We study the parity-breaking higher-curvature gravity theory of Chern-Simons (CS), using the Palatini formulation in which the metric and connection are taken to be independent fields. We first show that Palatini CS gravity leads to first-order derivative equations of motion and thus avoid the typical instabilities of CS gravity in the metric formalism. As an initial application, we analyze the cosmological propagation of gravitational waves (GWs) in Palatini CS gravity. We show that, due to parity breaking, the polarizations of GWs suffer two effects during propagation: amplitude birefringence (which changes the polarization ellipticity) and velocity birefringence (which rotates the polarization plane). While amplitude birefringence is known to be present in CS gravity in the metric formalism, velocity birefringence is not present in metric CS gravity, but now appears in Palatini CS due to the fact that left-handed and right-handed GW polarizations have a different dispersion relation. In the approximation of small deviations from General Relativity (GR), we do find however that velocity birefringence appears at least quadratically in the CS coupling parameter $α$, while amplitude birefringence appears linearly in $α$. This means that amplitude birefringence will be the most relevant effect in Palatini CS and hence this model will behave similarly to metric CS. We confirm this by applying current constraints on amplitude and velocity birefringence to Palatini CS, and showing that those from amplitude birefringence give the tightest bounds.