We investigate surface plasmon polaritons (SPPs) modes in palladium (Pd)-coated silver double nanowires by using the finite difference time domain (FDTD) method. Since Pd can absorb hydrogen (H\(_2\)) and converts to Pd-H, its permittivity is completely different from that of Pd-H, so the optical response of the system will be also change. The results of electric field distributions and propagation lengths of seven modes of the designed structure are obtained by calculations. For different modes, it is found that the propagation length increases with the increase of mode order. The propagation lengths of the same mode in the structure of Pd and Pd-H are respectively compared, and it is discovered that it is also different due to the change of dielectric constant before and after H\(_2\) absorption. The results show that in the double nanowire structure, the propagation lengths of structures with Pd are larger than the ones of structures with Pd-H for each mode. The distance of the two nanowires impacts on the coupling between the two nanowires. By changing the radius of Ag nanowire, the propagation lengths of the fundamental mode increase, while the ones of the harmonic modes decrease. There is an optimal thickness to make the propagation length longest for each mode by changing the thickness of Pd, which shows the competition between the dissipation of the structure and the coupling between SPPs in different layers. The structure we designed can be applied to the direction of the hydrogen sensor to realize the monitoring of its state when using hydrogen energy to ensure safety.