In a large saucepan, combine the onion, carrot, oil, and 1/4 teaspoon salt. Cover and cook over medium-low heat, stirring occasionally, until the vegetables are softened, 8 to 10 minutes. Stir in the garlic and thyme and cook until fragrant, about 30 seconds.
Stir in the farro and toast until fragrant, about 2 minutes. Stir in the broth and water and bring to a simmer. Reduce the heat to low and continue to simmer, stirring often, until the farro is tender, 20 to 25 minutes.
Stir in the parsley and lemon juice and season to taste with salt and pepper.