..what is a good way to boost the songs and decrease the between song crowd noise while maintaining some level of cohesiveness?
Parallel compression is not the tool you want for adjusting this.
What page describes can work if you set things correctly so that the limiter / compressor only works on the loud, between song portions and doesn't do much to the songs. But for what you want to do, I'd suggest manually drawing volume changes with the envelope tool- higher when he's playing and lower when when the loud crowd comes in. If all the loud parts are mostly between songs there shouldn't be too many events to make doing that overly difficult and its easy to make the change up and down happen as quickly or slowly as needed by adjusting the slope between points. Alternately you could achieve the same thing using fader automation if you prefer working that way and your software does it.
Why?
If I understand correctly the dynamic range during the songs themselves is not the problem and doing it this way doesn't change it at all. You'll preserve the existing dynamics within the music segments and and within the louder between song segments, but will make difference between them less which is what you want. Compression or limiting will change the sound of the song portions, even if set well. Standard compression reduces dynamics by working most on the loudest parts and transients. Parallel compression instead changes the quiet end of things- which in this case are the song portions. Your recording may or may not benefit from that, but it's a separate dynamics issue than the song/applause level differences.