A highly parallel time integration method is presented for calculating viscoelastic flows with the DEVSS-G/DG finite element discretization. The method is a synthesis of an operator splitting time integration method that decouples the calculation of the polymeric stress by solution of a hyperbolic constitutive equation from the evolution of the velocity and pressure fields by solution of a generalized Stokes problem. Both steps are performed in parallel. The discontinuous finite element discretization of the hyperbolic constitutive equation leads to highly-parallel element-by-element calculation of the stress at each time step. The Stokes-like problem is solved by using the BiCGStab Krylov iterative method implemented with the block complement and additive levels method (BCALM) preconditioner. The solution method is demonstrated for the calculation of two-dimensional (2D) flow of an Oldroyd-B fluid around an isolated cylinder confined between two parallel plates. These calculations use extremely fine finite elements and expose new features of the solution structure.