We propose a machine learning framework for identifying parameters with steady-state solutions, locating these solutions, and determining their linear stability in systems of ordinary differential equations and parameterized dynamical systems. Our method begins with the construction of target functions, which can be viewed as non-Mercer kernels, to identify parameters associated with steady-state solutions and evaluate their linear stability. To approximate the target function, we design a parameter-solution neural network (PSNN) that integrates a parameter neural network and a solution neural network. Efficient algorithms are developed to train the PSNN and to locate steady-state solutions. We also establish a theoretical framework for approximating the target function using our PSNN, leveraging neural network kernel decomposition. Numerical experiments demonstrate the effectiveness of our approach in locating solutions, identifying phase boundaries in the parameter space, and classifying the stability of these solutions. While the primary focus is on steady states of parameterized dynamical systems, the methodology is broadly applicable to finding solutions for parameterized nonlinear systems of algebraic equations. Additionally, we discuss ongoing extensions of this work.