Context. We previously investigated the stability threshold of the ion-ion acoustic instability (IIAI) in parameter regimes compatible with recent Parker Solar Probe (PSP) observations, in the presence of a Maxwellian electron distribution. We find that the observed parameters are close to the instability threshold, but IIAI requires a higher electron temperature than what is observed. Aims. As electron distributions in the solar wind present clear non-Maxwellian features, we investigated if deviations from the Maxwellian distribution could explain the observed IIAI. We address specifically the kappa ( ) and core-strahl distributions for the electrons. Methods. We performed analytical studies and kinetic simulations using a Vlasov-Poisson code in a parameter regime relevant to PSP observations. The simulated growth rates were validated against kinetic theory. Results. We show that the IIAI threshold changes in the presence of or core-strahl electron distributions, but not significantly. In the latter case, simulations confirm the expression of an e ective temperature for an equivalent Maxwellian electron distribution. Such an e ective temperature could simplify stability assessments of future observations.