I usually wait until the vine it is attached to starts to dry up. Perhaps others will tell you to pick earlier. If you pick later you won't make a mistake unless you wait until it becomes moldy on the bottom.

They turn yellow as the fully mature, and continue to get deeper color in storage.

You can pick for immediate use any time after they are fully sized and the rind is beginning to harden. But, in order to store well, they do need to stay on the vine until they are completely mature, the rinds very hard, and the seeds inside are fully mature and hard. Really, unless rotting or threatened by predators or weather, the longer they can stay on the vine, the better. Spaghetti squash is sort of intermediate storage variety - not the worst keeper, but not the best, generally lasting about 2 to 3 months for me.