Tunable broadband terahertz (THz) sources with power at the gigawatt level are desirable for many applications. A scheme to generate such THz emission by off-axially injecting a weakly relativistic ultrashort laser into a parabolic plasma channel is presented. By utilizing two-dimensional particle-in-cell simulation, it is demonstrated that there are two major physical mechanisms involved, i.e., linear mode conversion from laser wakefields and the electromagnetic waveguide mode excitation inside the plasma channel. The two radiation modes can lead to linearly polarized and radially polarized THz emissions, respectively, with distinct frequency spectra and spatial distributions. It is found that they predominate alternatively with the change of the plasma channel length. For a given plasma channel, one can switch the radiation modes by adjusting the injection position and the injection angle of the laser pulse. In particular, the radiation mode of the linear mode conversion can produce THz pulses with the peak amplitude of sub-GV cm −1 with the energy conversion efficiency ≈10 −3 , even though the peak power of the incident laser is just at the terawatt level. The scheme provides a powerful THz source with tunable intensity, spatial distributions, spectra, and polarizations by simply adjusting the injection conditions of incident laser pulses.