Parity Hamiltonian path problem Wikipedia says that Hamiltonian path problem is NPC, but Parity Hamiltonian path problem (i.e., is there an odd amount of hamiltonian path) is P. Does a reduction from, e.g., SAT, to HPP, unavoidably duplicate some solutions?
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.