Self-interacting dark matter (SIDM) can resolve the final parsec problem in supermassive black hole (SMBH) mergers by providing sufficient dynamical friction to overcome the energy loss barrier. This mechanism allows SMBH binaries to inspiral and merge within a few gigayears, as observed in gravitational wave (GW) data from pulsar timing arrays. The self-interaction cross section of dark matter (DM) must be of order cm²/g to enable this effect. For collisionless cold dark matter (CDM), the DM spike is disrupted, preventing the SMBHs from merging. In contrast, SIDM's isothermal core can absorb the energy from the SMBH orbits, enabling the merger. A realistic velocity dependence, such as that from a dark photon mediator, provides a good fit to the GW spectrum and allows a large enough core. This velocity dependence also addresses small-scale structure problems in CDM. The DM density profile is determined by the SMBH mass, with the spike profile depending on the cross section and velocity dependence. The SMBH merger dynamics show that dynamical friction from the DM spike can significantly reduce the inspiral time, especially for SIDM. The energy lost by the SMBH binary is compared to the gravitational binding energy of the DM spike to determine if it can be absorbed. For SIDM, the isothermal core acts as a reservoir, allowing the SMBHs to merge. The GW spectrum is softened at low frequencies due to DM friction, matching observations. The best-fit model for the GW signal is the massive mediator model, which provides a large core and steep spike. This model is compatible with small-scale structure constraints and matches the observed GW spectrum. The study shows that SIDM can resolve the final parsec problem and match the observed GW spectrum, while CDM cannot. The results suggest that gravitational wave signals from SMBH mergers can be used to probe dark matter microphysics.Self-interacting dark matter (SIDM) can resolve the final parsec problem in supermassive black hole (SMBH) mergers by providing sufficient dynamical friction to overcome the energy loss barrier. This mechanism allows SMBH binaries to inspiral and merge within a few gigayears, as observed in gravitational wave (GW) data from pulsar timing arrays. The self-interaction cross section of dark matter (DM) must be of order cm²/g to enable this effect. For collisionless cold dark matter (CDM), the DM spike is disrupted, preventing the SMBHs from merging. In contrast, SIDM's isothermal core can absorb the energy from the SMBH orbits, enabling the merger. A realistic velocity dependence, such as that from a dark photon mediator, provides a good fit to the GW spectrum and allows a large enough core. This velocity dependence also addresses small-scale structure problems in CDM. The DM density profile is determined by the SMBH mass, with the spike profile depending on the cross section and velocity dependence. The SMBH merger dynamics show that dynamical friction from the DM spike can significantly reduce the inspiral time, especially for SIDM. The energy lost by the SMBH binary is compared to the gravitational binding energy of the DM spike to determine if it can be absorbed. For SIDM, the isothermal core acts as a reservoir, allowing the SMBHs to merge. The GW spectrum is softened at low frequencies due to DM friction, matching observations. The best-fit model for the GW signal is the massive mediator model, which provides a large core and steep spike. This model is compatible with small-scale structure constraints and matches the observed GW spectrum. The study shows that SIDM can resolve the final parsec problem and match the observed GW spectrum, while CDM cannot. The results suggest that gravitational wave signals from SMBH mergers can be used to probe dark matter microphysics.