In a medium skillet, heat the 1/2 cup of olive oil. Add the capers in an even layer and cook over moderate heat until the capers start to sizzle, about 2 minutes; stand back, as the capers may spatter. Reduce the heat to moderately low and continue to cook until the capers are crisp, about 4 minutes longer. With a slotted spoon, transfer the capers to paper towels to drain.
Stir the anchovies and garlic into the olive oil in the skillet. Simmer over low heat until the garlic starts to turn golden, about 10 minutes. Remove from the heat.
Light a grill. Brush the zucchini and squash slices with olive oil and season with salt and pepper. Grill over high heat until they are just tender and nicely charred, about 2 minutes per side. Transfer the zucchini and squash to a platter.
Rewarm the anchovy dressing and add the lemon zest, lemon juice and minced parsley. Drizzle the warm bagna cauda over the grilled zucchini and squash, sprinkle with the fried capers and parsley leaves and serve.