No. The thing is, although it's a convention to use "dimension" as a synonym for "parallel universe/plane," that's actually a misnomer, or at best a shorthand. A dimension is a direction, an axis of motion, a degree of freedom in a space. If the universe has more than three dimensions, then there could be other 3D universal planes coexisting alongside ours through a fourth or higher dimension/direction -- analogously to how flat, 2D sheets can be stacked on top of each other in a third, up-down dimension. It's more accurate to say that these other planes can be accessed through other dimensions, that they're parallel to us within a higher-dimensional space, but sloppy usage leads to them being called "dimensions" in themselves.
So it's not the planes that are believed to be microscopic, it's the dimensions they're embedded in. Our universe and others may all be embedded in a higher-dimensional space, most of whose dimensions are compactified.
So how can they coexist side-by-side when the dimensions in which they're side-by-side are so tiny? Well, I'm not sure on this point, but I think the idea is that the extra dimensions within our own universe are compactified, but our universe may exist in a larger higher-dimensional space in which those dimensions aren't compactified. Again, let's reduce it to a 3D analogy. A strand of hair or a length of wire has very little extent in two of our universe's three dimensions -- it has a significant length, but its width and height are tiny, compactified. But it resides inside a larger universe where length, width, and height are all limitless in extent.
So what might be going on is that there are many parallel universes (maybe literally parallel in a geometric sense, alongside our universe in a higher dimension but never touching it) that have amounts of matter comparable to our own. If, say, 90-100 of them are close enough for us to feel their gravity, that could explain why we detect the gravitational effects of nearly 100 times as much mass as we can see. Though that's grossly oversimplified, I'm sure.
