Make aioli: In small bowl, stir together mayonnaise, yogurt, lemon juice, mustard, garlic, and paprika. Set aside.
In large saucepan, cover potatoes with salted water. Bring to a simmer; cook, partially covered, until fork-tender, 20 to 25 minutes. Drain, and quarter.
In a medium saucepan, cover carrots with salted water. Bring to a simmer; cook until tender, about 5 minutes. Transfer to a plate with a slotted spoon. Cook beans in the water until tender, 6 to 8 minutes. Transfer to plate. Cook shrimp until pink, 1 to 2 minutes. Drain.
Divide the aioli among 4 bowls. Arrange eggs, vegetables, and shrimp on plates. Garnish with parsley. Serve warm, with the aioli on the side.
This recipe is so easy and the aioli garlic mayonnaise is so tasty!