To dry herbs like parsley (I do the same with rosemary and oregano), I cut stems near the base, while the leaves are still young enough to have a good amount of oils. Choose healthy, undamaged stems and leaves. After washing and blotting dry all the stems/leaves, I let them air dry until there is no residual moisture on the leaves, about an hour depending on the weather. This helps prevent mold from forming while it dries in the bag.
Then, gather them up by the stems and put them into a clean paper bag. The stems should be at the top of the bag, with the leaves inside. Don't use too many stems in each bag, as there should be a good ratio of air to leaves in the bag. If you're using a small, lunched sized bag, use only about a half a dozen stems. If you have a larger bag, you can add a few more, but don't try to do too many at once. Make sure to use a bag big enough so that the leaves don't reach the bottom of the bag... you want there to be lots of room and air around the leaves in there!
Using about 18" of cotton string, wrap one end around the top of the bag, including the stems, so they are held snugly at the top of the bag. With the other end, make a loop so you can suspend the bag.
Punch some holes in your bag for air circulation. Hang the bag up in a room that has good air circulation (not a closet!) and check on it occasionally, starting at about 2 weeks. The leaves should be completely dried and crumble easily. (If there is any mold, discard the bunch.) I usually remove the leaves from the stems, breaking them up by hand before placing them in a jar for storage.
Enjoy with all your Italian dishes!