This paper presents a novel physics-constrained parallel network for nonlinear dynamical system identification. The fundamental concept is to employ implicit governing equations to guide neural network training, constraining the solution space and inducing interpretable models. Firstly, inspired by sparse regression methods, a sparse regression layer equipped with a function library is developed to characterize system nonlinearity. Secondly, a state-constrained parallel network architecture is constructed to enforce derivative relationships among state variables, constraining the outputs of three parallel subnetworks and reconstructing the full-state outputs under partially noisy state measurements. Finally, the sparse regression network layer is integrated with the state-constrained parallel network to form the physics-constrained parallel network, yielding full-state outputs and explicit closed-form dynamical formulations simultaneously. An alternate optimization method is developed to optimize the sparse regression network layer and the state-constrained parallel network sequentially, enhancing optimization efficiency. The term “physics-constrained” herein refers to the state dependency constraints and the residual loss derived from the learned governing equation via the sparse regression layer. Through this strategy, the proposed framework delivers a physically interpretable model for nonlinear dynamical systems from partial noisy measurements. Numerical simulations and experimental studies demonstrate the effectiveness, robustness, and applicability.