A parallel ladder is a situation in which the attacker can make two ladders on top of each other. The attacker's ladders are connected to each other and proceed in the same direction (both left to right or both right to left). Here is a typical example:
2nd and 4th rows
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. Indeed, if the attacker wants to, they have the option of only pushing the 4th row ladder (and ignoring the 2nd row ladder), or of only pushing the 2nd row ladder (and ignoring the 4th row ladder). Thus, both 2nd row ladder escapes and 4th row ladder escapes can be used to escape parallel 2nd-and-4th row ladders. However, there are some ladder escapes that work for parallel ladders, but not for individual ladders.
The best-known example of this is Tom's move, which escapes are parallel 2nd-and-4th row ladder without requiring any pre-existing pieces on the board. Tom's move only requires a certain amount of empty space. 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.
Even without a ladder escape, a parallel ladder is awkward to defend against and will often give an advantage to the attacker. For example, a parallel ladder gives the attacker good climbing opportunities. Also, a parallel ladder is no worse than the lower ladder plus a switchback threat.
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 climb:
At this point Red fails to connect. Is Red done for? No! Red can escape the parallel ladder 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 "*".
3rd and 5th rows
It is also possible to have a parallel ladder on the 3rd and 5th rows, such as this:
There are a few other things the defender can do, but all of them result either in the attacker connecting or getting a 2nd row ladder with switchback threat. This is discussed in more technical detail here.
There is a version of Tom's move for 3rd-and-5th row parallel ladders, but it requires a large amount of space.