Liquid neural networks (LNNs) are a class of continuous machine learning models in which the evolution of a hidden state is described by a system of ordinary differential equations. A significant advantage of LNNs is their time adaptability and the presence of strict theoretical guarantees of input-to-state stability (ISS), which ensures reliable operation under noisy, irregular, and undistributed input data. These properties make LNNs promising for application in control systems, robotics, and time series processing. Despite these advantages, the practical use of LNNs is limited by high computational complexity. Numerical methods for solving ordinary differential equations have a pronounced sequential nature, which complicates efficient parallelization of computations and leads to increased training and inference times, especially when implemented on GPUs and distributed computing platforms. This paper discusses a parallelization method for LNNs: Stable Parallel Liquid Recurrence (SPLR). The approach is based on representing the problem of computing the trajectory of a hidden state over a time interval as the solution of a system of nonlinear equations equivalent to the discretized dynamics of an LNN. An iterative fixed-point method, which allows parallel execution, is used to solve it. To maintain stability at each iteration, a stabilizing projection onto the parameter and state domain in which the ISS conditions are satisfied is introduced. Theoretical results are obtained establishing the convergence of the proposed iterative algorithm and the preservation of the input-state stability property upon transition to a parallel computing scheme. It is shown that the stabilizing projection prevents the accumulation of numerical errors and limits the growth of disturbances arising during parallel processing. Unlike existing methods for parallelizing recurrent computations, which are primarily focused on speedup, the SPLR method preserves the key advantage of LNNs: their robustness to input and numerical disturbances. The practical significance of this work lies in expanding the applicability of liquid neural networks in real-time systems and on embedded platforms with limited computing resources.