How to Store Fresh Herbs
If you're going to store anything leafy in a plastic bag, I wrap it first in a paper towel, then in the plastic bag, so none of the leaves touch the bag.  This prevents the issue where the outer leaves turn to goo.  (I'm not sure what the actual biological issue is ... moisture/condensation? poor respiration?)

Don't wash it before storage, as the extra moisture will cause it to rot faster.

I think I've gotten as long as a month out of flat leaf parsley, when I buried them in the back of my crisper drawer and forgot about them.  (of course, I wasn't constantly opening the bag as you would when using them, and I'm not sure exactly when I placed it in there, as I had forgotten about it).