Deep neural networks are not resilient to parameter corruptions: even a single-bitwise error in their parameters in memory can cause an accuracy drop of over 10%, and in the worst cases, up to 99%. This susceptibility poses great challenges in deploying models on computing platforms, where adversaries can induce bit-flips through software or bitwise corruptions may occur naturally. Most prior work addresses this issue with hardware or system-level approaches, such as integrating additional hardware components to verify a model's integrity at inference. However, these methods have not been widely deployed as they require infrastructure or platform-wide modifications. In this paper, we propose a new approach to addressing this issue: training models to be more resilient to bitwise corruptions to their parameters. Our approach, Hessian-aware training, promotes models with $flatter$ loss surfaces. We show that, while there have been training methods, designed to improve generalization through Hessian-based approaches, they do not enhance resilience to parameter corruptions. In contrast, models trained with our method demonstrate increased resilience to parameter corruptions, particularly with a 20$-$50% reduction in the number of bits whose individual flipping leads to a 90$-$100% accuracy drop. Moreover, we show the synergy between ours and existing hardware and system-level defenses.