# Magnetic parity and electric parity parts of solutions?

I'm currently reading the paper Conserved charges of the extended Bondi-Metzner-Sachs algebra by Flanagan and Nichols. In equation (2.15), the solution $$Y^A = D^A\chi + \epsilon^{AB}D_B\kappa$$ is written, where $$\chi$$ and $$\kappa$$ are $$l=1$$ spherical harmonics, and $$A, B$$ are the angular coordinates on the 2-sphere.

What I am confused about is that the paper says that the $$D^A\chi$$ term is electric parity, while the $$\epsilon^{AB}D_{B}\kappa$$ term is magnetic parity.

I thought that since $$l=1$$ spherical harmonics have odd parity, $$\chi, \kappa$$ should go to $$-\chi, -\kappa$$ under a parity transformation. And I'm not sure if it is supposed to depend on what coordinate system we are using, but if we are using the complex stereographic coordinates the paper later uses, $$(z, \bar{z})$$ with $$z = \cot(\theta/2)e^{i\phi}$$ in conventional spherical coordinates, then under parity transformations, we should have $$z\to -1/\bar{z}, \bar{z}\to -1/z$$. And then, $$D^A= g^{AB}D_B$$ (since $$g$$ should not change under parity transformation) should become $$(\partial_z, \partial_{\bar{z}}) = g^{AB}(\bar{z}^2\partial_{\bar{z}}, z^2\partial_z)$$, which seems to neither be positive nor negative parity.

Why is the solution mentioned an example of these parities?