The vase method is great. I do that with asparagus as well as herbs.
Right now, I have a "bouquet" of cinnamon basil in a dark spot on my kitchen counter. Basil seems to prefer not to be in the fridge. My whole kitchen and living area smell great.
If I have an extra amount of parsley or basil, as I do when I have it growing in my vegetable garden, I put it in the food processor and give it a spin. I then put the pureed herbs in a ziplock, mash it out real thin, seal and freeze. You can then break little pieces off when you need it, and the aroma and taste is as fresh as if you just picked it!
I haven't had a decent parsley crop for two years. Last year, it got washed out, and this year, I couldn't find any plants. It's easy to start from seed, but that must be done earlier, so I'm doing without except when Kim buys me a bunch.