This paper develops the variational principle of minimum "extended" dissipation for slow (low Reynolds number) flows of nematic liquids as described by the five parametric Leslie-Ericksen-Parodi (LEP) constitutive equations. It is shown that the Euler's equations for minimizer of the extended dissipative functional are the Stokes equations for the LEP fluid. When the molecular (including magnetic) field is absent, the extended dissipative functional coincides with the true dissipative functional, whose Euler equations are the Stokes equations for the Ericksen fluid.