Partial differential equations (PDEs) provide fundamental models for simulating complex phenomena in science and engineering, with applications ranging from fluid dynamics to materials science. Solving (parametric) PDEs efficiently remains a critical challenge due to their high-dimensional and computationally intensive nature. Recent advances in computational methods have highlighted neural networks (NNs) as promising discretization tools for addressing these challenges, leveraging their ability to serve as universal function approximators. Neural networks have demonstrated success in diverse fields, offering innovative approaches to regression, classification, and the solution of initial and boundary-value problems. Unlike traditional numerical methods, NNs represent functional manifolds with nonlinear approximation properties, facilitating the exploration of high-dimensional and complex solution spaces. Neural networks are increasingly used in applications such as solving high-dimensional linear and nonlinear PDEs, uncertainty quantification (UQ), optimization problems, and inverse problems. Additionally, Physics-Informed Neural Networks (PINNs) have been introduced to enforce PDE constraints by collocating the strong residual on training points, proving effective in forward and inverse problem simulations. However, challenges persist in terms of runtime efficiency, error control, the choice of loss formulations, control of integration errors, design of proper optimization tools, and understanding the mathematical principles underpinning NN-based methods. This mini-symposium highlights recent advances in scientific machine learning (SciML) and neural network methodologies for solving parametric PDEs, with emphasis on mathematical analyses and computational improvements. Topics of interest include convergence and stability properties, a posteriori error estimation, adaptive strategies for network architectures, optimization techniques, proper integration strategies for neural networks, and neural network technologies combined with other traditional numerical methods like finite elements.