Trim flowers off of broccoli. Cook in boiling water for 3-4 minutes. Drain water and run cold water over immediately, to stop the cooking process. Drain excess water.
Melt the butter in a large skillet over a medium-high heat. Add the onion and saute to a rich golden color. Add the celery and corrot and saute for another 5 minutes.
Add the zucchini and red bell pepper to the skillet and continue to saute over a medium-high heat until all the vegetables are tender and lightly colored ( approximately 10-20 minutes), depending on the size of the skillet. Add salt and black pepper to taste.
Add the broccoli to the skillet and saute for about 1 minute. Add the cream and cook, stirring occasionally until the cream has reduced by half then remove from heat and set aside.
Boil water in sauce pan, and then add 1 tablespoon of salt and the pasta to the boiling water in the saucepan, stirring.
When the pasta is cooked al dente, return the skillet with the sauce to a medium heat, drain the pasta and toss it with the sauce, adding the grated cheese and parsley. Serve at once.