'dimension' here is used loosely. In fact, 'parallel dimension' doesn't make sense as a term. There are two ways to render it consistent:
It really means an affine space, in which the three dimensions we have and the three dimensions it has are the same, but everything there is displaced in a different dimension we are not aware of. Thus, one additional unobservable dimension allows infinite 'parallel dimensions'. If these 'parallel dimensions' are to act independently and not simply become a four or more dimensional space, matter in these parallel dimensions must be constrained not to interact with anything which differs in that dimension -- under common circumstances, at least. This is a fundamental anisotropy of a parallel dimension scheme.
We are not really in a 3-dimensional space, e.g. as in String Theory, which suggests that about 10 additional dimensions exist. These dimensions do not suggest the existence of anything in those other dimensions other than what we have here because the matter in our world DOES interact in all dimensions. However, the other dimensions are cyclic, with the largest of the nonconventional ones being about 10^-15 meters long (give or take a few orders of magnitude). Something else on this scale is the atomic nucleus. Our matter is spread out in those dimensions, and we'd never know it. Can't visualize this? Okay. When you play pool (if you play pool), you normally consider the balls interacting with the other balls in two dimensions. However, there is a third dimension, height. But the balls all have essentially the same height, so it doesn't matter. Similarly, we all have the same 'height' in these extra dimensions. If we pushed ourselves as far apart as we could possibly be in these dimensions, we would have hardly changed distance at all, simply because the added distance is so tiny.