Preheat the oven to 350°F (180°C). Place the garlic in a jar with the olive oil to infuse.
Place the bell pepper on a baking tray and roast for about 25 minutes, turning over as needed, until the skin is blistered and charred. Place in a bowl, cover with an upturned plate and leave to sweat for a few minutes. Remove the charred skin, stem and seeds, then cut the flesh into thick strips.
Cut the eggplant and zucchini into ¼ inch (4–6 mm) slices, using a mandoline or sharp knife. Trim the woody ends of the asparagus spears.
Place a cast-iron grill pan on the stovetop over medium-high heat, or heat a barbecue grill plate to medium–high.
Working in batches, brush the vegetables on one side with the garlicky olive oil and place that side down on the grill. After a few minutes, brush the other side with olive oil, before turning over with tongs.
Arrange all the grilled vegetables on a serving platter. Serve scattered with salt and parsley and a drizzle of balsamic vinegar.
If not serving immediately, store the vegetables in a covered container, and only drizzle with vinegar just before serving.