Purslane

3 tablespoons olive oil
1 tablespoon fresh lemon juice
1 tablespoon finely chopped shallot
1/2 pound cherry tomatoes (preferably assorted heirloom varieties), halved or quartered if large
6 cups packed tender purslane sprigs and leaves (from a 1-pound bunch)
4 cups packed flat-leaf parsley leaves (from 2 large bunches)