I'll do it here, so it'll be useful to more ppl.
Parallel compression is when you mix a signal with a compressed copy of it. You can do any kind of parallel processing, but if there are any phase shifts or delays, these will cause phase issues. Some effects are all about these phase issues (phaser being the obvious example), but they're usually not something you want.
A compressor without any filters or delays doesn't cause phase issues, so you can mix its output with the original signal without any problems. Unlike regular compression, this mixed result will not have that same squashed or pumping sound, even if you compress it really hard (depending on the mix ratio). you can use multiple compressors in sequence too.
This is often done on a bus that's mixed in with the main signal. But it's so much easier when you can just turn a single knob in an effect you're using anyway.
As for the rest of the signal chain, I'll probably have an EQ in there, and it'll end with a limiter and some monitoring, but the key element of the chain is probably the multiband compressor. It lets me control how much space the lows get to take up. It's like a dynamics-sensitive EQ. I usually don't put any of the other output effects on (except the limiter) until I have the sound design and most of the arrangement in place, because I don't want to use the end of the chain to correct things I should correct earlier. If I can easily solve something with a touch of level automation, I do that rather than make the multiband compressor work overtime.