We systematically studied the influence of Pd underlayer on the magnetic properties of perpendicular [Co/Ni]₅ and [Ni/Co]₅ multilayers by time-resolved magneto-optical Kerr effect. We found that the saturated magnetic damping constant α₀ increases continuously with increasing the Pd thickness t_Pd, showing no linear correlation with the perpendicular magnetic anisotropy constant K_u for both series of samples. As compared to the Co/Ni sample with the same t_Pd, the Ni/Co film shows lower saturation magnetization, weaker K_u but larger α₀, which can be ascribed to the presence of more paramagnetic spins at the Pd/Ni interface due to the weak exchange coupling stiffness between Ni atoms. By analyzing extrinsic damping contributions from magnetic inhomogeneities, two-magnon scattering and spin pumping effects, the intrinsic damping of perpendicular [Ni/Co]₅ is quantitatively determined to be 0.023. This study should provide deep understanding and effective control of magnetic damping for designing high performance magnetic memory devices.