In this paper, we consider the problem of distributed parameter estimation in sensor networks. Each sensor makes successive observations of an unknown $d$-dimensional parameter, which might be subject to Gaussian random noises. The sensors aim to infer the true value of the unknown parameter by cooperating with each other. To this end, we first generalize the so-called dynamic regressor extension and mixing (DREM) algorithm to stochastic systems, with which the problem of estimating a $d$-dimensional vector parameter is transformed to that of $d$ scalar ones: one for each of the unknown parameters. For each of the scalar problem, both combine-then-adapt (CTA) and adapt-then-combine (ATC) diffusion-based estimation algorithms are given, where each sensor performs a combination step to fuse the local estimates in its in-neighborhood, alongside an adaptation step to process its streaming observations. Under weak conditions on network topology and excitation of regressors, we show that the proposed estimators guarantee that each sensor infers the true parameter, even if any individual of them cannot by itself. Specifically, it is required that the union of topologies over an interval with fixed length is strongly connected. Moreover, the sensors must collectively satisfy a cooperative persistent excitation (PE) condition, which relaxes the traditional PE condition. Numerical examples are finally provided to illustrate the established results.