Recent models of preferential acceleration and heating of heavy ions in the solar wind have invoked a resonant interaction with parallel-propagating ion cyclotron waves. However, these waves must have frequencies in the cold plasma stop band (&OHgr;a<&ohgr;<&ohgr;c) to acclerate alpha particles to speeds faster than the bulk plasma. Such waves cannot exist in a plasma containing cold alpha particles. We investigate solutions to the WKB warm plasma dispersion relation in a proton-alpha plasma to determine when parallel ion cyclotron waves can propagate at &ohgr;=&OHgr;a. We present examples of the behavior of the dispersion curves in this resonant regime for several type of plasma states appropriate to the fast solar wind. We find that ion cyclotron waves can propagate at &ohgr;=&OHgr;a for a bounded set of plasma parameters. However, use of the correct dispersion relation is likely to make resonant cyclotron acceleration of solar wind alpha less feasible as a mechanism to produce the observed preferential effects.