Soak both the clams and mussels in well salted water for at least an hour. Drain when ready to cook.
In a saucepan large enough to hold all the clams and mussels comfortably, sauté the garlic and parsley stems gently in a drizzle of olive oil, just until you can smell the garlic (which should not brown at all).
Add the clams first and then the mussels, along with a splash of white wine.
Cover and let the clams and mussels steam let them steam just until they open. Shake the saucepan from time to time to help things along and uncover to check on them every couple of minutes. As the bivalves open, remove them to a bowl to cool off.
When the clams and mussels have cooled enough to handle, gingerly remove the meat from their shells, using paring knife if need be to cut the ligaments that attach the meat to its shell. Rinse off any grit that may cling to the meat and, for the mussels, trim off any threads that might still be clinging to them.
Place the clams and mussel meat in a small bowl and, if you're not planning on using them right away, moisten them with a spoonful or two of their cooking juices.
When you're ready to use the clams and mussels, drain them and gently pat them dry with a paper towel.