Parallel compression is a production technique famously created in New York City studios. It usually is achieved by combing a dry and uncompressed sound with a heavily compressed sound. When this is done the new sound retains transient information while bringing up the harmonic content. Sometimes compression is too heavy or obvious and it's better to blend the original sound with the newly compressed sound in order to get the benefits of both.
Compression is like distortion or overdrive. They both flatten peaks but in different ways. You can achieve similar results by using a distortion effect instead of a compression effect. It works the same but usually gives more vibe and harmonics with an actual drive effect rather than a compressor which usually just deals with volume. The most cliche compressor for parallel compression is an 1176 which distorts and compresses when pushed hard.
Try your favorite distortion plugs and boxes in place of your compressor in your parallel chain.