A low-loss and high-bandwidth polymer parallel optical waveguide with graded-index (GI) rectangular cores is fabricated for high-speed and high-dense optical interconnections. We demonstrate that the near-parabolic index profile formed in the rectangular-shaped core GI waveguide exhibits superior properties similar to those of GI circular core waveguides we previously reported. In particular, we focus on the modal dispersion in the GI polymer waveguides with rectangular cores by showing experimental results. In this paper, the GI rectangular cores are fabricated using the preform method. However, conventional photo-lithography and imprinting processes are viable to fabricate a similar waveguiding structure, by which fabrication of a printed circuit board embedding this waveguide would become feasible.