Trustworthiness has become a key requirement for the deployment of artificial intelligence systems in safety-critical applications. Conventional evaluation metrics, such as accuracy and precision, fail to appropriately capture uncertainty or the reliability of model predictions, particularly under adversarial or degraded conditions. This paper introduces the Parallel Trust Assessment System (PaTAS), a framework for modeling and propagating trust in neural networks using Subjective Logic (SL). PaTAS operates in parallel with standard neural computation through Trust Nodes and Trust Functions that propagate input, parameter, and activation trust across the network. The framework defines a Parameter Trust Update mechanism to refine parameter reliability during training and an Inference-Path Trust Assessment (IPTA) method to compute instance-specific trust at inference. Experiments on real-world and adversarial datasets demonstrate that PaTAS produces interpretable, symmetric, and convergent trust estimates that complement accuracy and expose reliability gaps in poisoned, biased, or uncertain data scenarios. The results show that PaTAS effectively distinguishes between benign and adversarial inputs and identifies cases where model confidence diverges from actual reliability. By enabling transparent and quantifiable trust reasoning within neural architectures, PaTAS provides a foundation for evaluating model reliability across the AI lifecycle.