  Parametric oscillation occurs when the resonance frequency of an oscillator
is periodically modulated. Owing to time-reversal symmetry breaking in magnets,
nonreciprocal magnons can be parametrically excited when spatial-inversion
symmetry breaking is provided. This means that magnons with opposite
propagation directions have different amplitudes. Here we demonstrate switching
on and off the magnon parametric oscillation by reversing the external field
direction applied to a Y$_3$Fe$_5$O$_{12}$ micro-structured film. The result
originates from the nonreciprocity of surface mode magnons, leading to
field-direction dependence of the magnon accumulation under a nonuniform
microwave pumping. Our numerical calculation well reproduces the experimental
result.
