I usually keep fresh parsley, cilantro and basil in the fridge, so I reuse spaghetti (mason) jars to store them in water. The other day I came across Jess from cooksmarts and checked out her Youtube channel. Not only does she offer recipes (most aren't vegan), but also advice and techniques to make your kitchen experience that much easier.
In this video, she offers advice for keeping your herbs fresh for several weeks. You simply cut off the ends (like flowers), place into a jar with water, then cover with a small plastic bag (the same bag that came with the herbs) in the refrigerator.
Another popular technique:
Store herbs in oil from The Kitchn