Nonlinear magnetization dynamics are of great interest, being, for instance, leveraged for neuromorphic computing in spin-transfer torque nano-oscillators. Here, we demonstrate how to implement magnetoacoustics to reach this regime, using monochromatic (f saw = 450 MHz) surface acoustic waves traveling on a thin layer of (Ga,Mn)As. By careful tuning of the precession frequency to both f saw and 2 f saw using the magnetic field and temperature, we evidence clear signatures of a nonlinear magnetoacoustic response of the magnetic dynamics using the time-and space-resolved Kerr effect: (i) frequency and wave-vector doubling in time and space, respectively, (ii) quadratic (sublinear) evolution of the precession amplitude at 2 f saw (f saw) with acoustic amplitude, and (iii) resonance field shift. While (i) can be well reproduced by a parametric resonance model where nonlinearities arise solely from the SAW, we show that features (ii) and (iii) also involve intrinsic magnetic nonlinearities. Understanding the conditions leading to these nonlinearities will mean better control of the acoustic-wave-driven magnetization dynamics, in order to implement optimally the wave properties enabled by this approach.