Irreducible parabolic geometries are a family of geometric structures including conformal semi-Riemannian structures, projective structures, and many more. Automorphisms of these structures need not be linearizable around a fixed point, in contrast to semi-Riemannian isometries or affine transformations of a connection. I will present rigidity theorems for structures of this type admitting special nonlinearizable flows by automorphisms. A consequence will be that in many cases, if the geometry is not flat---that is, not locally equivalent to the homogeneous model---then automorphisms are determined by their 1-jet at a point.