Theory is developed to predict the growth and structure of ``sibling'' waves developing through triad resonant instability of a vertically confined mode-1 internal ``parent'' wave in uniform stratification including the influence of background rotation. For a sufficiently hydrostatic parent wave, two branches for growth of sibling waves are dominant. The branch with largest growth rate corresponds to sibling waves having frequencies much larger than that of the parent; the other branch corresponds to sibling waves having frequencies close to half the frequency of the parent. Numerical simulations show that sibling waves corresponding to the subharmonic branch appear in practice. In the absence of rotation, the sibling waves corresponding to this branch are predicted to have near-constant growth rate as their horizontal wavenumber increases. With rotation, however, the growth rate peaks at moderate wavenumber. In all cases, as confirmed by numerical simulations, the e-folding time for the growth of the sibling waves can be thousands of buoyancy periods for parent waves having amplitudes typical of realistic oceanic internal modes. In non-uniform stratification, the parent wave self-interacts immediately to force superharmonics. Nonetheless, numerical simulations with symmetric top-hat stratification show that triad resonant instability eventually emerges. Such emergence is not evident in simulations with stratification more representative of the ocean. The results suggest a reconsideration of the efficacy of parametric subharmonic instability in leading to the breakdown of low-mode internal tides in the ocean.