Differential equations are fundamental for modelling complex scientific and engineering processes. Operator learning offers a powerful framework for approximating the mapping from parametric inputs such as initial/boundary conditions or forcings to the corresponding solutions. However, purely data-driven operator learning methods such as the Laplace Neural Operator (LNO) typically require large training datasets and exhibit limited extrapolation capabilities. We propose the Physics-Informed Laplace Neural Operator (PILNO), which embeds the governing physical laws directly within the learning process. This physics-informed approach reduces data dependency, enabling effective learning in small-data regimes and enhancing generalization to out-of-distribution inputs where purely data-driven LNOs often fail. By leveraging both data (when available) and physics, PILNO offers a more robust and data-efficient pathway to learning solution operators. We demonstrate advantages of PILNO across diverse parametric differential equation problems, highlighting its improved data efficiency and extrapolation performance.