Nano-structuring can significantly modify the properties of materials. We demonstrate that size-dependent modification of the spin-wave spectra in magnetic nano-particles can affect not only linear, but also nonlinear magnetic response. The discretization of the spectrum removes the frequency degeneracy between the main excitation mode of a nano-particle and the higher spin-wave modes, having the lowest magnetic damping, and reduces the strength of multi-magnon relaxation processes. This reduction of magnon-magnon relaxation for the main excitation mode leads to a dramatic increase of its lifetime and amplitude, resulting in the intensification of all the nonlinear processes involving this mode. We demonstrate this experimentally on a two-dimensional array of permalloy nano-dots for the example of parametric generation of a sub-harmonic of an external microwave signal. The characteristic lifetime of this sub-harmonic is increased by two orders of magnitude compared to the case of a continuous magnetic film, where magnon-magnon relaxation limits the lifetime.