In a small bowl, whisk together 2 tablespoons olive oil, tomato paste, garlic, rosemary, ½ teaspoon salt, and ¼ teaspoon pepper until smooth.
Cut the potatoes (with the skin on) into ¼-inch thick slices.
Place the potatoes in a bowl and toss with the sauce until evenly coated.
Spread the potatoes out onto a large, unlined sheet pan in a single layer with minimal overlap.
Transfer to the oven and roast for 20 minutes.
Meanwhile, toss the sliced portobello and cremini mushrooms in the same bowl with the remaining 2 tablespoons olive oil, balsamic vinegar, remaining salt, and remaining pepper.
Remove the pan from the oven and flip the potato slices with metal tongs.
Scatter the mushrooms over and around the potatoes, keeping everything as close to a single layer as the pan allows.
Return the pan to the oven and roast for 20 to 25 minutes more, until the mushroom edges are deeply browned and curling, the potato edges are crisp and golden, and a paring knife slides into a potato slice with no resistance.
Sprinkle parsley and lemon zest over the mushroom potato bake before serving.