The aperture-coupled hemispherical dielectric resonator antenna (DRA) with a parasitic patch is studied rigorously. Using the Green's function approach, integral equations for the unknown patch and slot currents are formulated and solved using the method of moments. The theory is utilized to design a circularly polarized (CP) DRA and a wide-band linearly polarized (LP) DRA. In the former, the CP frequency and axial ratio (AR) can easily be controlled by the patch location and patch size, respectively, with the impedance matched by varying the slot length and microstrip stub length. It is important that the AR will not be affected when the input impedance is tuned, and the CP design is therefore greatly facilitated. For the wide-band LP antenna, a maximum bandwidth of 22% can be obtained, which is much wider than the previous bandwidth of 7.5% with no parasitic patches. Finally, the frequency-tuning characteristics of the proposed antenna are discussed. Since the parasitic patch can be applied to any DRAs, the method will find applications in practical DRA designs.