In this paper, we propose a probabilistic transmission scheme for distributed parameter estimation in wireless sensor networks. We assume that sensor observation noises are Gaussian distributed with non-identical statistics and the fusion center does not know the sensors' noise statistics. Each sensor employs a data rate to quantize its analog measurement that is a function of its signal-to-noise ratio (SNR). In order not to exceed the available capacity, for each possible data rate, the quantized sensor data are sent to the fusion center with a certain transmission probability. Under total bandwidth and network utilization constraints, we formulate an optimization problem to find the optimal transmission probabilities of each data rate by minimizing the inverse of the average Fisher information of the estimate. Under stringent availability of bandwidth, simulation results show that the proposed probabilistic transmission scheme outperforms the scheme where the total bandwidth is equally distributed among sensors. The optimal transmission probabilities are assigned in such a way that the sensors with high SNR have priority to transmit their data, and the mean squared estimation error is quite close to the case where all the sensors transmit at the maximum data rate.