In this study, we report a parallel algorithm for the infinite-size density matrix renormalization group (iDMRG) that is applicable to one-dimensional (1D) quantum systems with $\ell$-site periods, where $\ell$ is an even number. It combines Hida's iDMRG applied to random 1D spin systems with a variant of McCulloch's wavefunction prediction. This allows us to apply $\ell/2$ times of computational power to accelerate the investigation of multi-leg frustrated quantum systems in the thermodynamic limit, which is a challenging simulation. We performed benchmark calculations for a spin-1/2 Heisenberg model on a Kagome cylinder YC8 using the parallel iDMRG, and found that the proposed iDMRG was efficiently parallelized for shared memory and distributed memory systems, and provided such bulk physical quantities as total energy, bond strength on nearest neighbor spins, and spin--spin correlation functions and their correlation lengths without finite-size effects. Moreover, the variant of the wavefunction prediction sped up Lanczos methods in the parallel iDMRG by approximately three times.