We investigate the conditions under which parallel-propagating
Alfv\'en/ion-cyclotron waves are driven unstable by an isotropic ($T_{\perp
\alpha} = T_{\parallel\alpha}$) 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 $\simeq 5%$ of the proton number density and
the two species have similar thermal speeds, the instability requires that
$\beta_{\rm p} \gtrsim 1$, where $\beta_{\rm p}$ is the ratio of the proton
pressure to the magnetic pressure. For $1\lesssim \beta_{\mathrm p}\lesssim
12$, the minimum $U_{\alpha}$ needed to excite this instability ranges from
$0.7v_{\mathrm A}$ to $0.9v_{\mathrm A}$, where $v_{\mathrm A}$ is the Alfv\'en
speed. This threshold is smaller than the threshold of $\simeq 1.2v_{\mathrm
A}$ for the parallel magnetosonic instability, which was previously thought to
have the lowest threshold of the alpha-particle beam instabilities at
$\beta_{\mathrm p}\gtrsim 0.5$. We discuss the role of the parallel Alfv\'enic
drift instability for the evolution of the alpha-particle drift speed in the
solar wind. We also analyze measurements from the \emph{Wind} spacecraft's
Faraday cups and show that the $U_{\alpha}$ values measured in solar-wind
streams with $T_{\perp \alpha}\approx T_{\parallel\alpha}$ are approximately
bounded from above by the threshold of the parallel Alfv\'enic instability.