We consider parallel shear flows of an incompressible Newtonian fluid with a pressure-dependent viscosity. It is proved that pressure driven parallel flow in the plane is possible only if the dependence of the viscosity on the pressure is linear. In this case, a parallel flow exists for a pipe of arbitrary cross-section. For the special case of axisymmetric flow, we identify three cases for which pressure driven parallel flow exists. (C) 2003 Elsevier B.V. All rights reserved.