It is well-known that the computing power of natural and artificial neural networks arises from massive parallelism of simple "computing elements" [l]. Both natural neural networks and future molecular devices are "analog" computers subject to inaccuracies and some randomness, in contrast to present digital simulations. In this presentation, we show that trained artificial neural networks can be made of relatively inaccurate molecular components without adversely affecting their input-output behavior. The redundancy offered by parallelism in these networks overcomes inaccuracies in "computations" performed by any single component and yields accurate input-output behavior. This complements previous results on parallelism.