In a 5- to 6-quart pan over high heat, bring 3 quarts water to a boil.
Meanwhile, peel potatoes; cut into 3/4-inch cubes. Trim and discard ends from beans; remove strings. Cut beans diagonally into 2- to 3-inch lengths.
When water is boiling, add potatoes and pasta and cook, uncovered, stirring occasionally, for 8 minutes. Add beans and cook, stirring occasionally, until potatoes are tender when pierced and pasta and beans are barely tender to bite, 4 to 6 minutes longer. Drain, reserving 1 cup cooking liquid.
Return pasta mixture to pan and add pesto and 3/4 cup cooking liquid; mix gently. If desired, add more reserved cooking liquid to moisten. Gently stir in salt, pepper, and parmesan cheese to taste. Pour into a wide serving bowl or distribute evenly among plates.