The recent experimental discovery of $p$-wave magnetism opens new avenues for exploring momentum-dependent topological phases. However, whether $p$-wave magnetic proximity can generate higher-order boundary states remains largely unexplored, particularly when an odd in momentum $p$-wave channel induced by magnetic proximity coexists with a finite Zeeman-like perturbation. Here we investigate a quantum spin Hall insulator subjected to these two magnetic terms and show that the resulting higher-order topological phase arises when the Zeeman-like term shifts the helical Dirac crossing to finite momentum and the $p$-wave channel opens a gap at the shifted crossing. Specifically, the Zeeman-like term first shifts the helical Dirac crossing to finite momentum, enabling the $p$-wave channel to generate finite boundary masses. The resulting sign-changing boundary mass term creates domain walls that trap zero-energy corner states. For the family of model Hamiltonians parametrized by ${\varphi }_p$, the bound-state positions follow the orientation of the odd-parity form factor along curved boundaries. Our work establishes odd-parity magnetic proximity as a distinct route to higher-order topology in helical systems.

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• Topological edge modes
• Topological insulators
• Tight-binding model