In a mini food processor, pulse the parsley, garlic and 1 tbsp. EVOO into a chunky puree; season.
Season the potatoes. In a large nonstick skillet, heat 2 tbsp. EVOO over medium. Add half the potatoes and 5 of the thyme sprigs. Cook, partially covered, stirring occasionally, until the potatoes are browned and tender, 12 minutes. Remove from the skillet, transfer to a large bowl and discard the thyme sprigs. Repeat with 2 tbsp. EVOO and the remaining potatoes and thyme sprigs.
In the same skillet, heat the remaining 1 tbsp. EVOO. Add the mushrooms and shallots and cook, stirring occasionally, until the mushrooms are browned and the shallots are tender, about 6 minutes. Add the potatoes and cook until heated through, about 3 minutes. Stir in the parsley mixture.