Yes, you need parental permission to move if you are a minor. As a minor, you do not get to decide where you want to live. A custody order with a parenting plan is not invalid because your father moved and even if it was you would not get decide where you live.
I am not exactly sure what the situation is based on your question. Additional facts are needed. A modification of the custody order would seem to be required if you want to move back to Washington. You appear to be living in Minnesota, and want to live with one of your parents who live in Washington. I am not sure if your father moved from OR to WA or MN. I am not sure who has physical custody of you, but apparently he or she lives in Minnesota.