In a large pot, cover the fingerling potatoes with 2 inches of salted water and bring to a boil over high heat. Reduce the heat to moderate and cook until the potatoes are almost tender, about 8 minutes. Add the cauliflower florets and green beans to the pot and simmer until tender, about 5 minutes.
Meanwhile in a small skillet, heat the olive oil. Add the garlic and cook over low heat until golden brown, about 1 minute. Add the pimentón and a large pinch of salt and remove from the heat.
Drain the vegetables well and transfer to a serving bowl. Season with salt and drizzle with the garlic oil. Sprinkle with the vinegar, toss well and serve.