This work is motivated by the need to solve realistic problems with complex energy, space, and angle dependence, which requires parallel multigroup transport sweeps combined with efficient acceleration of the thermal upscattering. We present various iterative schemes based on the two-grid (TG) diffusion synthetic acceleration (DSA) method. In its original form, the TG method is used with the Gauss-Seidel iterative scheme over energy groups, which makes it impractical for parallel computation. We therefore formulate a Jacobi-style version. Furthermore, we propose a new scheme that reduces the overall number of transport sweeps by removing the need to fully converge the within-group iterations before the TG step. This becomes possible by adding an additional within-group DSA solve after each transport sweep. Fourier analyses are carried out to ascertain the effectiveness of the proposed scheme, with further corroboration from massively parallel numerical results from practical problem calculations. We discuss several implementation strategies of the new scheme, paying particular attention to the consequences on the overall efficiency of adding additional diffusion solves with a relatively low number of degrees of freedom per process.