Variational Data Assimilation (VDA) frameworks, particularly those using the adjoint technique, have long been the preferred approach for calibrating physical model parameters to align with observed data. However, these methods may appear complex and still computationally expensive for high-dimensional parameter identification. Recent advances in Machine Learning, especially Physics-Informed Neural Networks (PINNs), present exciting opportunities to address these challenges. This work revisits two PINN-based approaches for inverse problems, emphasizing their ability to infer high-dimensional physical parameters. The first, termed here Fully-Parameterized PINN, constructs a parameter-differentiable surrogate model through initial offline training, followed by rapid online parameter identification. This method treats physical parameters as NN inputs, making it prone to the curse of dimensionality. The second variant, called here Semi-Parameterized PINN (SP-PINN), integrates physical parameters as NN parameters, enabling efficient inference regardless of dimensionality via automatic differentiation. In this work, these methods are applied to the inference of spatially-distributed physical parameters (e.g., friction or infiltration coefficients) in flood models, a critical task for improving forecast accuracy. To evaluate their performance, several numerical experiments will be presented, including cases based on real-world data. In particular, SP-PINN is tested on a representative scenario for identifying a 1000-dimensional spatial friction parameter in a Shallow-Water model. Comparisons with more traditional VDA methods will also be shown, demonstrating the simplicity and efficiency of SP-PINN, and thus establishing it as a viable alternative in real-world parameter identification tasks.