Graph Neural Networks (GNNs) have emerged as powerful tools for nonlinear Model Order Reduction (MOR) of time-dependent parameterized Partial Differential Equations (PDEs). However, existing methodologies struggle to combine geometric inductive biases [1] with interpretable latent dynamics [2], overlooking dynamics-driven features or disregarding geometric information, respectively. In this work, we address this gap by introducing Latent Dynamic Graph Convolutional Network (LD-GCN), a purely data-driven, encoder-free architecture that learns a global, low-dimensional representation of dynamical systems conditioned on external inputs and/or parameters [3]. The temporal evolution is modeled in the latent space and advanced through time-stepping, allowing for time-extrapolation, and the resulting trajectories are consistently decoded onto geometrically parametrized domains using a GNN. Our framework enhances interpretability by enabling the analysis of latent trajectories and supports zero-shot prediction through interpolation in the latent space. The methodology is mathematically validated via a universal approximation theorem for encoder-free architectures, and numerically tested on complex computational mechanics problems involving physical and geometrical parameters, including the detection of bifurcating phenomena for Navier--Stokes equations. [1] Pichi, F., Moya, B., Hesthaven, J.S., 2024. \textit{A graph convolutional autoencoder approach to model order reduction for parametrized PDEs}. Journal of Computational Physics 501, 112762. https://doi.org/10.1016/j.jcp.2024.112762 [2] Regazzoni, F., Pagani, S., Salvador, M., Dedé, L., Quarteroni, A., 2024. \textit{Learning the intrinsic dynamics of spatio-temporal processes through Latent Dynamics Networks}. Nat Commun 15, 1834. https://doi.org/10.1038/s41467-024-45323-x [3] Tomada, L., Pichi, F., Rozza, G., 2026. \textit{Latent dynamics graph convolutional networks for model order reduction of parameterized time-dependent PDEs}. In preparation.