Parallel processing is simply to copy an audio signal and bias it with effects and then play it back mixed with the original, dry signal. By doing this, you’re sure to get a beefy sound.
Run the parallel channel through some compression, EQ it to accentuate certain frequencies, or use other effects to add character. (You can use an auxiliary bus to split the original sound to a parallel channel.) Now mix it with the original channel to keep the integrity of the original sound. Adjust the levels to get the effect that you what.
Look out for phase issues though. If your mixer (or instrument) offers to invert the phase, then this might solve the problem.
Parallel processing is a punchy technique with great effect. Use it to merge features of two (or more) sounds, and sculpt a new fat tone. E.g. try parallel compression on your drums and parallel distortion on your vocals or bass.