Continuous parallel mechanisms, characterized by slender and flexible members, present significant challenges for kinematic analysis due to the use of large-deformation theory and the inherent multiplicity of solutions to the position problem. This paper introduces an alternative forward-kinematics strategy that employs a feed-forward neural network trained on data generated by a Cosserat-rod physics-based numerical solver. The methodology is applied to a representative planar parallel linkage, using input features such as applied load magnitude, the specific rod affected, and boundary conditions to predict the deformed nodal coordinates of the mechanism. The trained neural network achieved high fidelity in replicating the linkage’s behavior, with a mean absolute error of approximately 3.04 mm and a coefficient of determination near unity on test cases not seen during training. These findings highlight the potential of the proposed data-driven approach for accurate deformation prediction and real-time position estimation in continuous parallel manipulators.