We consider distributed parameter estimation in energy-constrained wireless sensor networks, where limited energy is allowed to be used by all sensors at each task period. Thus there exists a tradeoff between the number of active sensors and the energy used by each active sensor to minimize the estimation MSE. To determine the optimal energy scheduling of each sensor, a concept of the equivalent unit-energy MSE function is introduced. Based on this concept, an optimal distributed estimation algorithm for homogeneous sensor networks and a quasi-optimal distributed estimation algorithm for heterogeneous sensor networks are proposed. Moreover, a theoretical non-achievable lower bound of the estimation MSE under the total energy constraint is proved and it is shown that our proposed algorithm is quasi-optimal and within a factor 2 of the theoretical lower bound. Simulation results also show that a significant reduction in the estimation MSE is achieved by the proposed method when compared with other uniform schemes.