In the present study, the mixed convection flow of nanofluid in a lid-driven parallelogramic porous enclosure subjected to a magnetic field is investigated numerically. Induced magnetic field is also considered, in terms of the magnetic potential, in solving the magnetohydrodynamic (MHD) flow and temperature equations. The Darcy-Brinkman-Forchheimer model with the Boussinesq approximation is adopted and the finite volume method based on SIMPLE algorithm is utilized to solve the governing equations with the appropriate boundary conditions in an orthogonal computational domain. The governing equations in a non-orthogonal physical domain are transformed into a computational domain in an orthogonal co-ordinate by co-ordinate transformations. It has been shown that the flow field and heat transfer are sensible greatly to the skew angle variation. Magnetic potential circulates through the parallelogramic porous enclosure with either high either magnetic Reynolds number or magnetic permeability of the nanofluid. Results also indicate that the influence of external magnetic field on fluid characteristics and heat transfer manifests variety fashions mainly depends on the effective area of the parallelogramic enclosure. Besides, the variation in rates of heat transfer while adding nanoparticles or applying magnetic field are affected to some extent by porous medium permeability and Richardson number.