Prepare the fish: Whisk together olive oil, lemon juice, parsley, garlic, basil, black pepper, and salt in a bowl and pour into a resealable plastic bag. Add tilapia fillets, coat with the marinade, squeeze out excess air, and seal the bag. Marinate in the refrigerator for 1 hour.
Meanwhile, make the salsa: Combine mango, bell pepper, jalapeño pepper, red onion, and cilantro in a bowl. Add lime and lemon juices and toss well. Season to taste with salt and pepper. Refrigerate until ready to serve.
Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Remove tilapia from the marinade and shake off excess. Discard remaining marinade. Grill fillets until fish is no longer translucent in the center and flakes easily with a fork, 3 to 4 minutes per side, depending on the thickness of the fillets.