It is not necessaryly the case that black holes connect two parallel universes. But instead, that if there are parallel universes (at least one), a black hole (the sink) in one universe can connect to another universe through a wormhole (the pipe). The exit of a black hole in another universe or in another region of our universe would be a white hole (the pipe exit). If there is a parallel universe the likelihood that there is another Earth (Sun, etc.) is zero. In order for them to exist there should be a huge number of universes (and even in this case the existence of a duplicate is not granted).
To imagine how two universes are connected imagine that our universe is a two-dimensional rubber planar sheet, and the paralle universe another one that is parallel to ours and separated by some distance. The third dimension is (a fourth dimension) not observable from within the universes, but can connect them: imagine that you press one sheet and deform it such that now the two sheets are conected trough a cilindrical rubber tube (made of "space"), which is analogous to the wormhole.