Herein, based on the TE₁ mode theory of the parallel-plate waveguide and assumption of approximately linearly polarized electric fields, we derive the expression of electric-field propagation of a cos-Gaussian beam in the parallel-plate waveguide. Under the paraxial approximation, the cos-Gaussian beam can be used as a fundamental solution for parallel-plate waveguides. The derived beam is tested by numerical simulation in copper parallel-plate waveguides at a frequency of 0.35 THz. Results show that the simulated amplitude and phase are in good agreement with those from the analytical expression of electric field. The deviation between the simulated phase and theoretical results at the z-axis and completeness of the electric-field divergence equation are also discussed.