It is important from an engineering aspect to understand drag-reducing viscoelastic flow, since comprehension of the drag-reduction mechanism may facilitate designs with better additive types and/or channel configurations. In this study, we investigated the attenuation of vortices in fully-developed turbulence of viscoelastic fluid between parallel planes by DNS (Direct Numerical Simulation). We discussed how the viscoelastic stress play a role in the reduction of the turbulent motions around vortices and how it influences the entire flow field. For determining the major effect of viscoelasticity, we classified three stages of vortex evolution and examined statistical data for each stage. We clarified the relation between the velocity field and viscoelastic stress field using a product, or a correlation, between vorticity and rotational viscoelastic stress. We found that the inhibition of turbulence would be caused by the following aspects: damping longitudinal vortex by viscoelastic stress over a wide range in the channel, and relaxing development of vortex by the stress in vortex growing state.