Iterative arrays (IAs) are a parallel computational model with a sequential processing of the input. They are one-dimensional arrays of interacting identical deterministic finite automata. In this paper, realtime-IAs with sublinear space bounds are used to accept formal languages. The existence of an infinite proper hierarchy of space complexity classes between logarithmic and linear space bounds is proved. Some decidability questions on logarithmically space bounded realtime-IAs are investigated. Furthermore, an optimal space lower bound for non-regular language recognition on realtime-IAs is shown.