We consider parametric instabilities of a circularly polarized pump Alfv¿n wave, which propagates parallel to the ambient magnetic field; the daughter waves are also parallel-propagating. We follow Hollweg et al. (1993) and consider several new instabilities that owe their existence to the presence of streaming alpha particles. One of the new instabilities is similar to the familiar decay instability, but the daughter waves are a forward going alpha sound wave and a backward going Alfv¿n wave. The growth rate of this instability is usually small if the alpha abundance is small. The other three new instabilities occur at high frequencies and small wavelengths. We find that the new instability which involves the proton cyclotron wave and alpha sound (i.e., the (+f, &agr;s) mode) can be the fastest growing instability if &bgr;≈1. However, if &bgr; is small, then the instability which can compete with the decay instability is the (+f, -&agr;) instability, which involves both the proton and alpha cyclotron resonances, but the pump wave must have low frequency and large amplitude. These instabilities may be a means of heating and accelerating alpha particles in the solar wind, but this claim is unproven until a fully kinetic study is carried out. ¿ American Geophysical Union 1994