Parametric message passing (MP) is a promising technique that provides reliable marginal probability distributions for distributed cooperative positioning (DCP) based on factor graphs (FG), while maintaining minimal computational complexity. However, conventional parametric MP-based DCP methods may fail to converge in dense wireless networks due to numerous short loops on FG. Additionally, the use of inappropriate message approximation techniques can lead to increased sensitivity to initial values and significantly slower convergence rates. To address the challenging DCP problem modeled by a loopy FG, we propose an effective graph neural network enhanced fast convergent parametric MP (GNN--FCPMP) method. We first employ Chebyshev polynomials to approximate the nonlinear terms present in the FG-based spatio-temporal messages. This technique facilitates the derivation of globally precise, closed-form representations for each message transmitted across the FG. Then, the parametric representations of spatial messages are meticulously refined through data-driven graph neural networks (GNNs). Conclusively, by performing inference on the FG, we derive more accurate closed-form expressions for the a posteriori distributions of node positions. Numerical results substantiate the capability of GNN--FCPMP to significantly enhance positioning accuracy within wireless networks characterized by high-density loops and ensure rapid convergence.