Place potatoes in a large saucepan, cover with cold water, and bring to a boil. Reduce heat to low. Simmer until just cooked, about 15 minutes; drain; cut into halves or quarters.
In a large bowl, combine corn, onion, celery, and cooked potatoes. In a jar, with a tight-fitting lid, combine oil, vinegar, salt, pepper, and thyme. Shake well. Pour over salad and stir to coat well. Add parsley and stir again.