Place potatoes in a large saucepan; add water to cover. Bring to a boil. Reduce heat; cook, uncovered, until tender, 8-10 minutes.
Meanwhile, chop green onions, reserving white portions. Add green portions to a blender. Add avocados, parsley, mayonnaise, tarragon, capers and seasoned salt. Cover and process until blended, scraping down the side as needed.
Drain potatoes; transfer to a large bowl. Add celery, white portions of green onions, and dressing; toss to coat. Refrigerate, covered, for at least 1 hour. Before serving, top with radishes and additional parsley.