Prior studies of the planar shear layer using an approach combining the Parabolized Stability Equations (PSE) and the acoustic analogy method have suggested that it can capture both the dynamics and acoustics of subsonic and supersonic instability modes. In this investigation we apply the hybrid PSE-acoustic analogy method to the study of subsonic and supersonic round jets. Using nonlinear PSE, a variety of jet conditions are simulated, including subsonic and supersonic, heated and unheated, and axisymmetric- and nonaxisymmetric instabilities of jets. When compared against the corresponding DNS data, we find the results agree well with prior observations of the shear layer. For supersonic jets with an instability wave phase velocity Mph > 1, the PSE method is able to capture the both the near-field hydrodynamics and the far-field acoustic radiation. For subsonic jets with Mph < 1, the PSE method can only capture the near-field accurately. However, by combining acoustic source terms calculated from PSE with an acoustic analogy method, reasonable predictions of the acoustic field can still be made.