We investigate properties of a solar wind-like plasma, including a secondary alpha particle population exhibiting a
parallel temperature anisotropy with respect to the background magnetic field, using linear and quasi-linear
predictions and by means of one-dimensional hybrid simulations. We show that anisotropic alpha particles can
drive a parallel fire hose instability analogous to that generated by protons, but that, remarkably, can also be
triggered when the parallel plasma beta of alpha particles is below unity. The wave activity generated by the alpha
anisotropy affects the evolution of the more abundant protons, leading to their anisotropic heating. When both ion
species have sufficient parallel anisotropies, both of them can drive the instability, and we observe the generation
of two distinct peaks in the spectra of the fluctuations, with longer wavelengths associated to alphas and shorter
ones to protons. If a non-zero relative drift is present, the unstable modes propagate preferentially in the direction
of the drift associated with the unstable species. The generated waves scatter particles and reduce their temperature
anisotropy to a marginally stable state, and, moreover, they significantly reduce the relative drift between the two
ion populations. The coexistence of modes excited by both species leads to saturation of the plasma in distinct
regions of the beta/anisotropy parameter space for protons and alpha particles, in good agreement with in situ solar
wind observations. Our results confirm that fire hose instabilities are likely at work in the solar wind and limit the
anisotropy of different ion species in the plasma.