  We investigate the generation of the frequency components at the second-order
sidebands based on a spinning resonator containing a degenerate optical
parametric amplifier (OPA). We show an OPA driven by different pumping
frequencies inside a cavity can enhance and modulate the amplitude of the
second-order sideband with different influences. We find that both the
second-order sideband amplitude and its associated group delay sensitively
depend on the nonlinear gain of the OPA, the phase of the field driving the
OPA, the rotation speed of the resonator, and the incident direction of the
input fields. Tuning the pumping frequency of the OPA can remain the
localization of the maximum value of the sideband efficiency and nonreciprocal
behavior due to the optical Sagnac effect, which also can adjust the linewidth
of the suppressive window of the second-order sideband. Furthermore, we extend
the study of second-order sideband to the non-Markovian bath which consists of
a collection of infinite oscillators (bosonic photonic modes). We illustrate
the second-order sidebands in a spinning resonator exhibit a transition from
the non-Markovian to Markovian regime by controlling environmental spectral
width. \textbf{We also study the influences of the decay from the non-Markovian
environment coupling to an external reservoir on the efficiency of second-order
upper sidebands.} This indicates a promising new way to enhance or steer
optomechanically induced transparency devices in nonlinear optical cavities and
provides potential applications for precision measurement, optical
communications, and quantum sensing.
