Parallel or ‘New York’ compression is simply the concept of splitting a signal at source into two parallel paths, compressing one while leaving the other unscathed. Once you blend the two back together, you should, in theory, be combining the benefits of compression while retaining the air and cleanliness of the source.
However, in truth, the technique tends to be used to take advantage of the colour a compressor can add, so in many cases you’ll be pushing the compressor harder and then blending the unprocessed signal back in with that. You’ll typically see it used for processing transient heavy sounds such as kicks and snares, and also for fattening up lacklustre vocals. Get it right and the results will differ from simply using the compressor on its own. But it can be a fiddly technique, and it’s very dependent on the level balance between the two signals.