When I pick herbs from my garden I sometimes take more than I need and put the excess into a tiny bouquet in a bud vase, like this thyme, and enjoy looking at it for a while before I use it within the next few days.
First take the parsley or cilantro out of the bag and get rid of the twist tie or rubber band that holds the bunch together and often cuts into the stems.
Then wash it well and pick out any bad leaves or stalks. Set any good but short stalks to one side. I put the bad parts in the compost bucket and the short stems in a small (snack-size) zip-lock bag that goes in the refrigerator to be used first. Below you see (part of) the larger bunch (in this case cilantro) at the very top of the photo, the shorter stalks on the left, and on the right the bruised or yellowing bits I'm going to throw away.
Put the big bunch in a glass of water, making sure there are no leaves under the water where they will quickly rot, just as you would with a bouquet of flowers. If you chop a half-inch or so off the bottom of the stems they'll be able to take up water more efficiently and stay fresher, again, just like a floral bouquet.
Place a plastic bag over the whole thing, glass and bouquet, pull up the excess plastic so the bag fits tight around the glass and fasten the wad of plastic with a twist tie so it will stay that way. Then set the glass on a shelf in your refrigerator where the parsley or cilantro will stay fresh much longer than if you just left it in the bag it came in, with a twist tie or rubber band cutting off the individual stems' circulation. This way you can use what you need, as you need it, while the rest remains appetizing.