Heat 1 tablespoon oil in a nonstick skillet over medium heat. Cook chorizo, turning often, until a meat thermometer inserted in center reads 165°F, 5 to 7 minutes. Let cool slightly. Cut into 24 pieces.
Combine parsley, cilantro, vinegar, jalapeño, garlic, salt, and remaining 2 tablespoons oil in a mini food processor; process until almost smooth.
Cut a slit down center of each date. Place 1 sausage piece in middle of each. Mold date around sausage. Arrange dates on platter; spoon herb sauce evenly over top.