Heat oil in medium saucepan on medium heat. Add onions and celery; cook 3 min. or until crisp-tender, stirring occasionally. Add mushrooms; cook 3 min. or until mushrooms are tender, stirring occasionally. Add broth; stir. Bring to boil. Stir in rice; cover. Reduce heat to medium-low; simmer 5 min. Remove from heat; let stand 5 min. Add walnuts and parsley; mix lightly.