  We investigate (local) automorphisms of parabolic geometries that generalize
geodesic symmetries. We show that many types of parabolic geometries admit at
most one generalized geodesic symmetry at a point with non-zero harmonic
curvature. Moreover, we show that if there is exactly one symmetry at each
point, then the parabolic geometry is a generalization of an affine (locally)
symmetric space.
