Wash the zucchini and yellow squash thoroughly under cold running water. Pat them dry with paper towels. Trim the ends and slice lengthwise into ¼-inch planks, ensuring uniform thickness for even cooking.
In a small bowl, whisk together the olive oil, minced garlic, Italian herbs, smoked paprika, salt, and pepper until well combined.
Place the sliced vegetables in a large shallow dish or zip-top bag. Pour the marinade over them, gently tossing to ensure even coating. Allow them to marinate for at least 15 minutes at room temperature, or up to 2 hours in the refrigerator.
Heat your grill to medium-high heat (approximately 400-425°F). Clean the grates thoroughly and oil them lightly using a paper towel dipped in vegetable oil and held with tongs.
Place the marinated vegetable slices directly on the grill grates, arranging them perpendicular to the grates. Grill for 3-4 minutes per side until they develop nice char marks and become tender but not mushy.
Transfer the grilled vegetables to a serving platter. Drizzle with fresh lemon juice while still hot and sprinkle with grated Parmesan if using.