In a recent paper, Hollweg et al. (1993) studied the parametric decay of Alfv¿n waves in high-speed solar wind streams. Following this analysis, we consider the nonlinear decay of left-hand-polarized ion cyclotron waves. It is shown that in a solar wind type plasma composed of electrons, protons, and alpha particles drifting relative to the protons, both branches of the dispersion relation of the circularly polarized waves can be excited by observed thermal anisotropies (Gomberoff and Elgueta, 1991). Guided by this analysis, the parametric decay of each branch of the dispersion relation is discussed. It is shown that the presence of drifting alpha particles introduces new wave couplings in the system that lead to new instabilities. Some of these instabilities involve sound waves supported essentially by the alpha particles, which, due to Landau damping, can be very efficient in the energization of alpha particles. Other instabilities involve ordinary sound waves that can lead to proton heating. A modulational instability that involves two electromagnetic daughters is also found. We have also found that a strong pump can force decays of modes that do not satisfy the resonance conditions when the pump intensity is vanishingly small. Finally, it is shown that both branches of the dispersion relation, particularly the branch close to the Doppler-shifted alpha particle resonance, are highly unstable even for small intensities of the pump wave.