This paper introduces a novel self-aligning parallel plate fixture (SAPP) for rotational rheometers. The ring-shaped shearing surface of this fixture is held on a low friction single contact point bearing and uses hydrodynamic lubrication forces in order to maintain the parallelism of the freely tilting surfaces over a full rotation. The optimized parallelism of the plates enables to conduct tribological measurements of low frictional stress between the shearing surface materials and a fluid at normal loads down to 1.3 kPa. Limited only by the degree of non-flatness of the surfaces, the new fixture can determine boundary lubrication sliding frictions within 10% and down to angular velocities of 400 µrad/s. In a controlled gap mode this setup reaches a gap error of 3.4 µm which enables to reliably conduct rheological measurements down to absolute gaps of the parallel plates of 10 µm and to reach high shear rates up to 10^5 s^-1.