In Minnesota, can a father terminate his parental rights if both parties are in agreement?

If you had read the sticky note on this issue or some of the hundreds of threads in this forum about this issue you would have had your answer.

Doesn't matter whether you are in Minn or Ark or anywhere in the US. A TPR has to be granted by a court and courts generally don't grant TPRs except to clear the way for an adoption or if the parent represents a danger to the child.