In a large stock pot, add enough cold water to cover the potatoes over an 1 inch. Add 1 tbsp. salt. Bring to a boil and cook until very tender, about 12 to 15 minutes.
Meanwhile, in a small saucepan, heat 2/3 cup olive oil on medium heat. Add the garlic, rosemary, lemon zest, crushed red pepper, and 1/4 tsp. salt, stirring occasionally, for 2 minutes.
Drain the potatoes. Push the potatoes through a potato ricer back into the pot, or return the potatoes to the pot and mash with a potato masher. Strain the oil mixture into the potatoes and add 1 cup warm vegetable broth. Add seasoning from oil, salt, and pepper to taste. Transfer the potatoes to a bowl, drizzle with oil, and sprinkle with the parsley.