The best example I can give is with a snare drum I recently recorded. The top head was an Evans, Genera Dry, which is a coated head designed to reduce ring. Unfortunately, it also made the whole drum sound darker. I was struggling to get some life into the track. The raw track was too thin for EQ alone so, it definitely needed compression.
With straight compression, the snare seemed to lose all of its "crack" or "pop." (That is because there is a lot of mid and high frequency content within the initial transient peak of the waveform. Fast compression tends to reduce that transient which really changes the sound of the drum.) It ended up louder and fatter in the lows, but with far less life in the mids. It just wouldn't cut through the mix.
Parallel compression was a good work-around in that situation. I made a duplicate copy of the track and compressed only one of them. Then mixed them back together. The compressed part added fullness to the track. The uncompressed part still preserved the original voice of the head.