In this paper, we maximize the parameterized energy efficiency (EEP) of a multi-channel fading communication link that simultaneously transmits data corresponding to delay-sensitive and delay-insensitive applications subject to a constraint on the minimum effective capacity (EC) of delay-sensitive traffic. EEP, in b/s/Hz, is defined as the difference between the spectral efficiency (SE) and the transmit power scaled by a parameter which represents the penalty on the transmit power. We provide an optimal power adaptation policy to this problem which is either given by the global optimum, if feasible, or given by the boundary point where the objective function intersects the constraint. Further, we apportion the total transmit power into delay-sensitive power, which is the minimum power required to meet the EC constraint and delay-insensitive power, which is the residual power used to maximize the EEP of the link. Simulation results show that delay-sensitive power is an exponentially increasing function of minimum EC while delay-insensitive power decays to zero for higher values of minimum EC. Further, delay-insensitive power increases while delay-sensitive power exponentially decreases with number of subchannels in the system.