Perovskite quantum dots (PQDs) emerged as a promising class of material for applications in lighting devices, including light emitting diodes and lasers. In this work, we explore nonlinear absorption properties of PQDs showing the spectral signatures and the size dependence of their two-photon absorption (2PA) cross-section, which can reach values higher than 106 GM. The large 2PA cross section allows for low threshold two-photon induced amplified spontaneous emission (ASE), which can be as low as 1.6 mJ/cm2. We also show that the ASE properties are strongly dependent on the nanomaterial size, and that the ASE threshold, in terms of the average number of excitons, decreases for smaller PQDs. Investigating the PQDs biexciton binding energy, we observe strong correlation between the increasing on the biexciton binding energy and the decreasing on the ASE threshold, suggesting that ASE in PQDs is a biexciton-assisted process.