In this paper, the power-allocation problem is solved in the context of maximizing the energy efficiency (EE) of orthogonal frequency-division multiplexing (OFDM) signaling over a two-way amplify-and-forward (AF) relay network with two sources and one relay. The EE is defined as the ratio of the system throughput over the total power consumption incorporating the transmission power, the fixed circuit power, and the dynamic circuit power, which is rate dependent. In the EE maximization (EEM) problem, the efficiency values of the power amplifiers (PAs) of the three terminals can differ from one another. It is shown that EE is a quasi-concave function of the transmission power. Based on this result, we use Dinkelbach's method for fractional programming and propose a two-step allocation (TSA) algorithm to solve the inner loop of Dinkelbach's method. Simulation results demonstrate that the proposed scheme can efficiently maximize the EE.