Parametric instabilities in the interaction between a cold-electron beam, a static or electromagnetic pump, and a scattered (signal) electromagnetic wave are studied. The interaction schemes are classified according to the mutual sense of propagation of the waves and the beam. The unified analysis yields the known convective and absolute slow-plasma-wave instabilities [forward- and backward-wave free-electron lasers (FEL)]. In addition, we identify the possibility of convective and absolute fast-plasma-wave instabilities. Analysis and numerical computation of gain and oscillation in both known and new instability mechanisms are performed in the framework of a general model beyond the range of previously reported studies. A general approximate threshold condition expression for all the possible absolute instabilities is given. All of these instabilities may appear in practical schemes like FEL oscillators as parasitic effects. Their relative importance is considered by means of numerical examples.