Cook pasta in a large saucepan of boiling, salted water according to packet instructions. Drain, then return to pan.
Meanwhile, heat oil in a large, deep frying pan over medium-high heat. Add onion and carrot, then cook, stirring, for 5 minutes or until softened. Add mince and cook, breaking up any lumps with a wooden spoon, for 4 minutes or until browned. Add garlic, paste and lentils, then cook, stirring, for 1 minute or until fragrant.
Add tomatoes and 1/2 cup water. Bring to the boil. Reduce heat to low and simmer, uncovered, for 10 minutes or until sauce thickens. Stir in spinach, season and add sauce and half the parsley to pasta. Toss to combine.