The GP is correct. Only great circles on a 2-sphere are geodesics. Curves of constant latitude, except for the equator, are not. I'm not sure why you reference

The GP is correct. Only great circles on a 2-sphere are geodesics. Curves of constant latitude, except for the equator, are not. I'm not sure why you referenced the Wikipedia article you did, since it supports what the GP says and contradicts what you say.