Pour olive oil into a 12-inch frying pan (with sides at least 2 in. tall) or 6- to 8-quart pan over medium heat. When warm, add crabs in the shell and cook, stirring occasionally, until juices leak into pan, 5 to 7 minutes. With tongs, transfer crab pieces to a large serving bowl; cover and keep warm in a 200° oven.
Add tomatoes (including juice), garlic, and basil to pan. With a wooden spoon, scrape any crab bits from bottom of pan. Cover and bring to a boil over high heat, stirring occasionally. Boil gently, uncovered, stirring occasionally and crushing tomatoes with a potato masher or spoon and reducing heat as sauce thickens, until sauce is thick and reduced to about 4 1/2 cups, about 30 minutes. Remove from heat. Spoon about a third of the sauce over crabs in the shell; cover loosely and return to oven.
About 15 minutes before sauce is done, in a 6- to 8-quart pan over high heat, bring 3 to 4 quarts water to a boil. Add pasta, stir to separate, and cook just until barely tender to bite, 3 to 4 minutes for dried angel hair pasta, about 2 minutes for fresh linguine. Drain.
Over low heat, add the shelled cooked crab to remaining tomato sauce; stir occasionally until hot, 1 to 2 minutes. Add salt and pepper to taste. Pour drained pasta into sauce and mix well.
Mound pasta in a wide, shallow serving bowl. Remove crab from oven and sprinkle both crab and pasta with parsley.