Add the lentils, carrot, garlic, olive oil, tomato paste, and salt to a large saucepan. Using the empty lentil can to measure, add 2 filled cans worth of water. Bring to a boil over medium-high heat and turn the heat down to low. Cover and simmer for 30 minutes.
After 30 minutes, uncover the saucepan and add the pasta pieces. Cook for an additional 5 minutes or until the pasta is al dente.
Serve immediately in individual pasta bowls and top with chopped parsley and grated parmesan, if desired.