We propose a minimal lattice model for two-dimensional class DIII superconductors with C-2-protected higher-order topology. Although this class of superconductors cannot be topologically characterized by symmetry eigenvalues at high-symmetry momenta, we propose a simple Wannier-orbital-based real-space diagnosis to unambiguously capture the corresponding higher-order topology. We further identify and characterize a variety of conventional topological phases in our minimal model, including a weak topological superconductor and a nodal topological superconductor with chiral-symmetry protection. The disorder effect is also systematically studied to demonstrate the robustness of higher-order bulk-boundary correspondence. Our theory lays the groundwork for predicting and diagnosing C-2-protected higher-order topology in class DIII superconductors.