We report on the parametric excitation of magnetic precession modes in nanodisks using a parallel pumping configuration. The excitations are detected using a ferromagnetic resonance force microscopy method, and the parallel-pumped spectra reveal nonlinear characteristics including instability thresholds and multiple, narrow, sawtooth-shaped resonances. These characteristics are in accord with analytical theory and micromagnetic modeling results. Modeled mode profiles of the excitations show that higher-order standing spin-wave modes with both even and odd symmetries are excited under parallel pumping.