Heat barbecue to medium heat. Heat oil in large skillet on medium-high heat. Add mushrooms; cook 3 to 5 min. or until softened and golden brown, stirring frequently and adding thyme and garlic to the skillet for the last minute. Stir in pasta sauce and milk; simmer on medium-low heat 3 to 5 min. or until heated through, stirring frequently. Stir in parsley. Remove from heat.