Physics-informed Neural Networks (PINN), embedding partial physics through minimizing residuals of ordinary/partial differential equations (ODEs/PDEs), show promise in reducing the reliance on extensive datasets for training by incorporating physical principles. However, their limitation in providing only point estimates hinders real-life applications where uncertainties are unavoidable. Addressing this, Bayesian Physics Informed Neural Networks (BPINN) with Hamiltonian Monte Carlo (HMC) optimization are employed here for uncertainty quantification in parameter and state estimation. BPINN, known for its versatility, provides interpretable solutions and robust parameter identification, making it particularly valuable in ocean engineering, where addressing significant uncertainties in system estimation poses a major challenge. This study applies BPINN to predict motion and identify system parameters in offshore structures, focusing on mooring lines. Using a simplified dynamics model, the research aims to predict the displacement of the platform surge and reveal the parameters of the system related to the degradation of the mooring line. Incorporating uncertainty quantification enhances the prediction reliability for real-world applications prone to environmental noise and uncertainties.