We define four families of geometries with as point graph the graph - or its complement - of all elliptic hyperplanes of a given parabolic quadric in any finite 6-dimensional projective space, where adjacency is given by intersecting in a tangent 4-space. One of the classes consists of semi-partial geometries constructed in J.A. Thas [SPG-reguli and semipartial geometries, Adv. Geom. 1 (2001) 229-244], for which our approach yields a new construction, more directly linked to the split Cayley hexagon. Our main results determine the complete automorphism groups of all these geometries. (c) 2005 Elsevier Ltd. All rights reserved.