Parity Hamiltonian path problem
As far as we know, there is no polynomial-time algorithm for HPP. Put differently, there is no known polynomial-time reduction from any NP-hard problem with the properties that you require to Parity-HPP.

If such a reduction did exist, we could reduce any NP-hard problem A to Parity-HPP (in polynomial time) and then solve A using the polynomial-time algorithm that you claim.