Standard physics-informed approaches for sparse parameter discovery often fail when the initial hypothesis is topologically disjoint from observed data due to significant model misspecification. This spatial misalignment causes a gradient vanishing pathology, where local differential operators are unable to propagate sensitivity information from sparse sensors to the unknown parameters. To resolve this, we propose the Physically-Relaxed PINN (PR-PINN), a homotopy optimization framework that temporarily introduces augmented residual operators via artificial viscosity and data nudging. These relaxation mechanisms broaden the effective support of the surrogate model to re-establish gradient flow, subsequently annealing toward strict physical constraints for fine-tuning. Experimental results on convection-diffusion dynamics demonstrate that PR-PINN successfully recovers critical source parameters in regimes where baseline methods stagnate, reducing estimation error by two orders of magnitude.