
Well, the clue is in the word 'parallel': basically you're taking a processed signal, along with the unprocessed original, and mixing them together. In this way, hopefully, you can combine the best of both worlds. It's generally applied to compression, because for most other types of processing it's not really needed; there's no point mixing a clean signal with something that has a large EQ boost - you might as well just have a small EQ boost in the first place. With compression, however, your signal is generally so radically altered, that there's not (always) a simple way to rein it in. Parallel compression is also sometimes known as New York compression.
The Basics

Techniques
A magic bullet, then? Well, not quite. It can still be a heavy-handed technique and is suitable for certain types of music more than others. If you fancy yourself as the next Steve Angello then go right ahead - slam that compressed signal into the mix. And then compress it again just to be sure. But for other sounds, where a degree of subtlety is required, you'll need to be careful that you don't lose what was good in the first place. Parallel compression is sometimes even referred to as 'upwards compression' - because instead of turning down the loud bits, the end result is that you turn up the quiet bits. Either way, you reduce the dynamic range, so bear it in mind.
I've assumed, here, that you'll be mixing in the box. This removes certain potential pitfalls; if you're rocking the hardware look then you'll know that by sending a signal out to a compressor, a delay is introduced, so mixing this back in may induce phase issues. Fortunately, for computer-based heads, any recent DAW - Logic 7 onwards, for instance - will have delay compensation built in; so just going through the basic parallel compression
process won't introduce phasing, in itself. There are still plenty of ways to do so though, so be careful. If you EQ your compressed signal, then an EQ can alter the phase of a signal, relative to itself. You can use a linear phase EQ, which should sort the issue - but really, it's best to get all that stuff out of the way before you get a compressor in there.
It's also easy to overdo any technique in the quest for more punch and louder mixdowns - so remember to check yourself as you go along.
So, that's the thinking behind it - let's have a look at how to set something up. For the sake of simplicity, I'll suggest that you take a full drum loop sample; maybe some live-sounding drums so that you'll be able to hear the effect most clearly (you could, of course, use your drum buss). Take a 'send' from this channel, send it to a buss, and put a compressor on it. Then go for some fairly vigorous processing; set your attack to 0ms, release something like 500ms, give it a ratio of 10:1 or more, and a low threshold. In other words, crush the life out of it - make sure to solo it, and you'll find it sounds horrible. Now, pull this fader all the way down (so you're hearing only the original), and then slowly bring it back up. What you should hear is that the drums sound beefier, with more weight, but without losing too much in the way of transient punch. At some point the compressed signal will start to overwhelm the clean - this is where you should back it off. Remember, if you can hear what's been done, then it's probably too much.
A magic bullet, then? Well, not quite. It can still be a heavy-handed technique and is suitable for certain types of music more than others. If you fancy yourself as the next Steve Angello then go right ahead - slam that compressed signal into the mix. And then compress it again just to be sure. But for other sounds, where a degree of subtlety is required, you'll need to be careful that you don't lose what was good in the first place. Parallel compression is sometimes even referred to as 'upwards compression' - because instead of turning down the loud bits, the end result is that you turn up the quiet bits. Either way, you reduce the dynamic range, so bear it in mind.
Pitfalls

process won't introduce phasing, in itself. There are still plenty of ways to do so though, so be careful. If you EQ your compressed signal, then an EQ can alter the phase of a signal, relative to itself. You can use a linear phase EQ, which should sort the issue - but really, it's best to get all that stuff out of the way before you get a compressor in there.
Keep it simple!
It's also easy to overdo any technique in the quest for more punch and louder mixdowns - so remember to check yourself as you go along.
This was a pretty technical article, but try it a couple of times and you'll find it all makes sense pretty quickly - you'll be giving your precious mixes more weight and clarity in no time. So - are you ready to bring the New York approach to your mixes?