Parallel optical interconnections which replace metallic transmission lines with optical fibers or free space channels provide high throughput, easy system integration, and low latency. These systems are used in making multiprocessor based supercomputers, telecommunication exchange switches and terminals, optical information processors and computers. A first-order model for the decrease in coupling efficiency between elements of two linear arrays of a free-space, parallel optical interconnect owing to misalignments or offsets in packaging is developed. Such an array interconnect consists of an array of optical sources, such as, optical fibers or VCSELs and an array of photo-receptors, such as, optical fibers, micromirrors or photodetectors. The coupling efficiency between source and receptor elements is modeled in terms of the sizes of the array elements, inter-element spacing and distances. The coupling efficiency is subject to degrading influence of six varieties of random offsets, which may occur during the alignment, and fixing of the two arrays in a package. We then determine first order approximations of the effects of these offsets. Our paper presents simple analytical formulas useful for a quick design of array-based parallel optical system packages and estimation of overall system performances. The formulas developed are useful for design and packaging of any optoelectronic processing system.