Many industrial coating fluids, particularly battery electrode slurries, exhibit complex non-Newtonian behaviors such as shear thinning and yield stress. Accurately simulating these flows is critical for process stability. Since flow within the coating bead is largely rectilinear, the 1D viscocapillary model—derived via the lubrication approximation—serves as an accurate and efficient surrogate for the full 2D flow, providing reliable estimates of flow fields and pressure drops. Deep learning offers a promising avenue to parametrically simulate this 1D model. However, purely data-driven supervised learning performs poorly due to the curse of dimensionality; even if supplied with millions of data points from a Finite Difference Method (FDM) solver, the training data remains sparse relative to the vast parameter space of material properties and operating conditions. In this work, we employ Physics-Informed Neural Networks (PINNs) to solve the parametric 1D viscocapillary problem with non-Newtonian rheology, incorporating shear-thinning and yield-stress effects. By directly embedding the governing lubrication equations into the loss function, the network learns the solution manifold without relying on pre-computed reference data. Validation against a standard FDM solver demonstrates that the PINN achieves a relative error of approximately 5% across the parameter space, highlighting its potential for rapid, data-free process optimization in non-Newtonian slot-coating processes.