A wave coupling formalism for magnetohydrodynamic (MHD) waves in a non-uniform background flow is used to study parametric instabilities of the large-amplitude, circularly polarized, simple plane Alfvén wave in one Cartesian space dimension. The large-amplitude Alfvén wave (the pump wave) is regarded as the background flow, and the seven small-amplitude MHD waves (the backward and forward fast and slow magnetoacoustic waves, the backward and forward Alfvén waves, and the entropy wave) interact with the pump wave via wave coupling coefficients that depend on the gradients and time dependence of the background flow. The dispersion equation for the waves D(k,ω) = 0 obtained from the wave coupling equations reduces to that obtained by previous authors. The general solution of the initial value problem for the waves is obtained by Fourier and Laplace transforms. The dispersion function D(k,ω) is a fifth-order polynomial in both the wavenumber k and the frequency ω. The regions of instability and the neutral stability curves are determined. Instabilities that arise from solving the dispersion equation D(k,ω) = 0, both in the form ω = ω(k), where k is real, and in the form k = k(ω), where ω is real, are investigated. The instabilities depend parametrically on the pump wave amplitude and on the plasma beta. The wave interaction equations are also studied from the perspective of a single master wave equation, with multiple wave modes, and with a source term due to the entropy wave. The wave hierarchies for short- and long-wavelength waves of the master wave equation are used to discuss wave stability. Expanding the dispersion equation for the different long-wavelength eigenmodes about k = 0 yields either the linearized Korteweg–deVries equation or the Schrödinger equation as the generic wave equation at long-wavelengths. The corresponding short-wavelength wave equations are also obtained. Initial value problems for the wave interaction equations are investigated. An inspection of the double-root solutions of the dispersion equation for k, satisfying the equations D(k,ω) = 0 and ∂D(k,ω) = ∂k = 0 and pinch point analysis shows that the solutions of the wave interaction equations for hump or pulse-like initial data develop an absolute instability. Fourier solutions and asymptotic analysis are used to study the absolute instability.