  We generalize the concept of affine locally symmetric spaces for parabolic
geometries. We discuss mainly $|1|$--graded geometries and we show some
restrictions on their curvature coming from the existence of symmetries. We use
the theory of Weyl structures to discuss more interesting $|1|$--graded
geometries which can carry a symmetry in a point with nonzero curvature. More
concretely, we discuss the number of different symmetries which can exist at
the point with nonzero curvature.
