"Street parking is always allowed"
"Not always! Three years ago there was a marathon that went through this street and you weren't allowed to park here that morning."
This exception is so specific and obscure that it "proves" (in a casual conversational sense) that "you can always park here" is a good rule. Not all exceptions prove the rule: if the exception is "except on weekends and holidays and overnight", that's so significant and obvious that it actually disproves the rule.
It was commonly mis-used to mean “eh that just a minor exception that you should ignore.” But it’s been mis-used so much that now it has both meaning.