A process specific to the dynamics of the heavy impurities in turbulent tokamak plasmas is found and analyzed. We show that the parallel stochastic acceleration is strongly coupled to the perpendicular transport and generates a radial pinch velocity. The interaction is produced with the hidden drifts, a quasi-coherent component of the motion that consists of a pair of average radial velocities in opposite directions. The parallel acceleration breaks this symmetry and yields a radial average velocity that can be in the inward or outward direction. The pinch is generated in three-dimensional turbulence, in the presence of a poloidal average velocity. It is significant only for heavy, high Z ions. The transport of the tungsten ions is analyzed in the frame of the minimal test particle model that yields this pinch mechanism. We use a semi-analytical method and numerical simulations. The scaling laws of the pinch velocity and of the diffusion coefficient are found and analyzed in order to drive a clear physical image of these non-linear effects. We conclude that the pinch produced by the non-linear interaction of the parallel accelerated motion with the perpendicular transport is rather strong for the heavy impurities.