In large frying pan, add butter and sausage links, fry until browned on the outside and done through, cooking over medium-low heat with lid on and turning as needed. Remove to a plate or cutting board.
Do not drain butter from pan, and add celery through garlic, cover and simmer til tender, (adding sausage back in after slicing as you go along), over medium-low heat, stirring every 5 minutes or so. (About 10 minutes depending on thickness of veggie slices.)
Remove lid, raise heat, and simmer, (gently stirring occasionally), at least 5 more minutes to cook out some of the liquid. You can remove any extra that you don't want with a spoon, too, just so that the sauce doesn't get too watered down.
Cook noodles in boiling salted water according to package directions, drain. Toss with olive oil so that they don't stick together.
While noodles are cooking, add sauce to veggie and sausage pan and stir, then simmer til heated through, uncovered. Then cover to keep warm while waiting for the pasta.
Stir in cooked and drained pasta. Garnish with parmesan and parsley, and enjoy!