Bring a small pot of water to a low boil, then add the garlic and simmer for 20 minutes. Let cool, then drain, peel and mash into a paste. Season with sea salt.
Bring a few inches of water to a boil in a large pot and season with salt. Fill a large bowl with ice water. Cut the vegetables into similar shapes and sizes but keep them separate. Blanch the vegetables, one kind at a time, until tender-crisp. Delicate vegetables like thin beans will take 1 to 2 minutes, firmer vegetables like squash will take 3 minutes. After each batch, transfer them to the ice bath to cool using a spider or a small strainer. Drain well.
Heat the EVOO in a small pot over medium heat. Add the anchovies and stir until they break down and melt into the oil. Add the heavy cream, butter and garlic paste, and season with salt and pepper. Cook, stirring, until thickened. Let the sauce cool, then transfer to an airtight container and refrigerate overnight (or longer).
Bring a large pot of water to a boil for the pasta and season with salt. Add the pasta and cook until al dente, 7 to 8 minutes. Add the vegetables in the last 2 minutes of cooking to reheat them. Reheat the bagna cauda sauce in a saucepan over medium-low heat.
Toss the pasta and vegetables with the sauce and the parsley. Season with salt and pepper.
You can blanch the vegetables ahead of time. Store them in a resealable plastic bag in the refrigerator.