Fill a large pot with lightly salted water and bring to a rolling boil over high heat. Stir in the linguine, and return to a boil. Cook, uncovered, stirring occasionally, until the pasta is cooked through but still firm to the bite, about 11 minutes; drain.
Return pasta to the pot to keep warm.
Meanwhile, melt the butter with olive oil in a large skillet over medium heat. Cook and stir the garlic, minced clams, and chopped clams for 5 minutes.
Add the reserved clam juice and cook for a few more minutes.
In a separate bowl, whisk the cornstarch with part of the bottled clam juice. Stir the cornstarch mixture into the skillet. Adjust the thickness of the sauce, if needed, by whisking a little more cornstarch with more bottled clam juice, and adding to the sauce.
Pour the remaining bottled clam juice into the skillet. Stir in the mushroom pieces, grated zucchini, cayenne pepper, and salt. Cook over medium heat, stirring occasionally, until zucchini is tender, about 5 minutes.
Stir in the parsley; remove from heat.
Pour the clam sauce over the drained linguine. Mix well; serve with Parmesan cheese.