Essential Killing fields of parabolic geometries Andreas CapKarin Melnick 5337Cartan geometryparabolic geometryalmost Grassmannian structuresCR structures We study vector fields generating a local flow by automorphisms of a parabolic geometry with \emph{higher-order fixed points}. We develop general tools extending the techniques of [T. Nagano and T. Ochiai, \textit{On compact Riemannian manifolds admitting essential projective transformations}, J. Fac. Sci. Univ. Tokyo Sect. IA Math. \textbf{33} (1986), no. 2, 233--246], [Ch. Frances, \textit{Local dynamics of conformal vector fields}, Geom. Dedicata \textbf{158} (2012), 35--59], and [Ch. Frances and K. Melnick, \textit{Formes normales pour les champs conformes pseudo-riemanniens}, Bull. SMF, 49 pp., to appear], and we apply them to almost-Grassmannian, almost-quaternionic, and contact parabolic geometries, including CR structures. We obtain descriptions of the possible dynamics of such flows near the fixed point and strong restrictions on the curvature; in some cases, we can show vanishing of the curvature on a non-empty open set. Deriving consequences for a specific geometry entails evaluating purely algebraic and representation-theoretic criteria in the model homogeneous space. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.5166 10.1512/iumj.2013.62.5166 en Indiana Univ. Math. J. 62 (2013) 1917 - 1953 state-of-the-art mathematics http://iumj.org/access/