A large amplitude Trivelpiece–Gould (TG) mode, in a strongly magnetized beam–plasma system, parametrically couples to a beam space charge mode and a TG mode sideband. The density perturbation associated with the beam mode couples with the electron oscillatory velocity, due to the pump wave, to produce a nonlinear current, driving the sideband. The pump and the sideband waves exert a ponderomotive force on the electrons with a component parallel to the ambient magnetic field, driving the beam mode. For a pump wave having k0·v0b0/ω0 < 0, where ω0, k0 are the frequency and the wave number of the pump, and v0b0 is the beam velocity, the sideband is frequency upshifted. At low beam density (Compton regime) the growth rate of the parametric instability scales as two-thirds power of the pump amplitude, and one-third power of beam density. In the Raman regime, the growth rate scales as half power of beam density and linearly with pump amplitude. The background plasma has a destabilizing role on the instability.