Make pesto: Place 1 tablespoon of the oil and garlic in a small skillet over low heat. When mixture begins to sizzle, let cook for 30 seconds, then transfer to a food processor. Add remaining 3 tablespoons oil, spinach, parsley, capers, hempseed, lemon zest, lemon juice, salt, and pepper; blend. (Makes 3/4 cup.)
Make zoodles: With a spiralizer or julienne peeler, slice zucchini into noodles. Warm 1 tablespoon of the oil in a large skillet. Add zucchini and sprinkle with 1/2 teaspoon salt; sauté until tender, about 1 minute. Transfer to a large bowl. Add a few tablespoons pesto; gently toss. (Cover and refrigerate leftover pesto for another use.)
Warm remaining 1 tablespoon oil in same skillet over medium heat. Add tomatoes; season with salt and pepper. Cook, shaking pan often, until tomatoes soften and begin to burst, 5 to 10 minutes. Add 3 to 4 tablespoons water partway through to incorporate browned bits from bottom of pan.
Top zoodles with tomatoes and Parmesan, if desired.