Heat the olive oil in a large pot over medium heat. Add the onion and cook for 5 minutes until it softens.
Stir in the garlic and cook for 1 minute until fragrant.
Add the mushrooms and carrots. Cook for 8 minutes, stirring now and then, until the mushrooms release their liquid and start to brown.
Pour in the farro, vegetable broth, thyme, rosemary, salt, and pepper. Bring the mixture to a boil.
Reduce the heat to low, cover the pot, and simmer for 30 minutes until the farro is tender.
Stir in half the parsley. Taste and adjust salt or pepper if needed. Serve the stew hot with the remaining parsley sprinkled on top.