A parallel ladder is a situation in which the attacker can make two ladders on top of each other.
2nd and 4th rows
Consider the following position with Red to play.
All of Red's pieces form a connected group. This group is connected to the top. At the bottom, Red has a second row ladder with no possible ladder escape on the left. The potential escapes on the right are inadequate. For example, suppose Red breaks the ladder at e9 and then tries to zipper:
At this point Red fails to connect. Is Red done for? No! Red can create a sufficient escape by making use of a parallel ladder, and essentially using Tom's move. Red plays like this:
Note that all of Blue's moves are forced. If Blue moves anywhere but 4, Red will easily connect to the edge. 3 and 7 are connected to the bottom edge by Edge template IV2b, so that 8 is also forced. Now Red is connected by double threat at the two cells marked "*".
A parallel ladder on the 2nd and 4th rows is a situation such as the following, with Red to move. The two red stones must be connected to the top edge (although the connection is not shown here). Red has the option of pushing the 2nd row ladder or the 4th row ladder:
The first essential point is that a parallel ladder can be pushed. If Red pushes on the 4th row, Blue does not have the option to yield, or else Blue will lose immediately.
Thus, Blue has no option but to push the ladder. Then Red can push the 2nd row ladder as well.
Note that pushing a parallel ladder only works if the 4th row ladder is "ahead" of the 2nd row ladder. Once the 2nd row ladder has caught up, it is too late to push on the 4th row, as Blue can then yield, resulting in an ordinary 3rd row ladder.
The second essential point is that a parallel ladder is stronger than either a 2nd row ladder or a 4th row ladder individually. The best-known way of escaping a parallel ladder is by using Tom's move, or a variation thereof, as shown in the example above. Tom's move only requires a certain amount of empty space, and does not require any pre-existing Red pieces. Even if there is not enough space before the ladder to perform Tom's move, a parallel ladder is awkward to defend against and will often give an advantage to Red.
There also exist other example (besides Tom's move) of ladder escapes that work for parallel ladders, but not for individual ladders. This is discussed in more detail here.
Here is an example of a parallel ladder that helps Red to connect, even though there is not enough space for Tom's move:
Red starts by pushing both ladders, then breaks the 2nd row ladder at 13. Note that every blue move is forced.
Red then moves upwards and eventually connects to the top edge by a double threat.
3rd and 5th rows
It is possible to use this trick off from one row farther back; i.e. with ladders on the third and fifth row but this occurs far less frequently and one has to examine some additional defensive possibilities. Consider the following position.
Red has just played e6 trying the parallel ladder trick. With the closer ladder on the second row, we saw that Blue was forced to respond with the parallel ladder play e7. But here Blue has two additional possibilities e8 and c9 (the only other possibility where Red doesn't have a way to force his group to connect to the bottom is c10. But Red can respond with f8 and now Blue has nothing better than e7, g6).
e8 yields a second row ladder after d8, e7, c8, c10, d9. The play c9 also leads to a second row ladder after the likely f7, f8, e8 (d9 is met by e7) d10. In the latter case, Red could again try the parallel ladder trick by playing g7. Of course, the presence of other pieces in the area can change the possibilities.