In order to describe the observed behavior of single motor proteins moving along linear molecular tracks, a class of stochastic models is studied which recognizes the possibility of parallel biochemical pathways. Extending the theoretical analysis of Derrida [J. Stat. Phys. 31, 433 (1983)], exact results are derived for the velocity and dispersion of a discrete one-dimensional kinetic model which consists of two parallel chains of N states and M states, respectively, with arbitrary forward and backward rates. Generalizations of this approach for g >2 parallel chains models are briefly sketched. These results and other properties of parallel-chain kinetic models are illustrated by various examples.