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.