We investigate the conditions under which parallel-propagating Alfven/ion-cyclotron waves are driven unstable by an isotropic (T sub(perpendicular alpha ) = T sub(|| alpha )) population of alpha particles drifting parallel to the magnetic field at an average speed U alpha with respect to the protons. We derive an approximate analytic condition for the minimum value of U alpha needed to excite this instability and refine this result using numerical solutions to the hot-plasma dispersion relation. When the alpha-particle number density is Asymptotically = to5% of the proton number density and the two species have similar thermal speeds, the instability requires that beta sub(p) > ~ 1, where beta sub(p) is the ratio of the proton pressure to the magnetic pressure. For 1 <, ~ beta sub(p) <, ~ 12, the minimum U sub( alpha ) needed to excite this instability ranges from 0.7v sub(A) to 0.9v sub(A), where v sub(A) is the Alfven speed. This threshold is smaller than the threshold of 1.2v sub(A) for the parallel magnetosonic instability, which was previously thought to have the lowest threshold of the alpha-particle beam instabilities at beta sub(p) > ~ 0.5. We discuss the role of the parallel Alfvenic drift instability for the evolution of the alpha-particle drift speed in the solar wind. We also analyze measurements from the Wind spacecraft's Faraday cups and show that the U sub( alpha ) values measured in solar-wind streams with T sub(perpendicular alpha ) approximately T sub(|| alpha ) are approximately bounded from above by the threshold of the parallel Alfvenic instability.