In standard braneworld models of cosmology, parallel universes can be thought of as separate branes evolving in a higher dimension space. So, for example, in such a set-up our universe would be a 3-brane (3 extended spatial dimensions). We can imagine a 4th spatial dimension orthogonal to the 3 dimensions of our brane. We could then in principle fill the 4th dimension with many other 3-branes (think of stacking cd's on a spool -- the cd's are 3-branes and the spool axis is the 4th orthogonal dimension). If each 3-brane could support a universe, then this construct would provide an example of parallel universes.
To answer your first question, the 'bulk' is the term used to describe the dimensions orthogonal to the brane of interest. The dimensionality depends on the model.