Toss together mushrooms, balsamic vinegar, 2 tbsp. olive oil, garlic and 1/2 tsp. each salt and pepper. Let stand for 5 minutes.
Stir together cilantro, parsley, jalapeno, shallot, red wine vinegar, remaining oil, salt and pepper. Fold in avocado.
Preheat grill to medium-high heat; grease grate well. Grill mushrooms for 4 to 5 minutes per side or until grill-marked and tender. Let stand for 2 minutes; slice thinly. Warm tortillas according to package directions.
Add spinach to center of each tortilla, leaving 1-inch border all around. Top with mushrooms, avocado chimichurri and tomato. Fold up bottom of tortilla over filling; fold in sides and roll up tightly starting from the bottom.