Does Christ's reign cancel out God the Father and God the Spirit?

No. Christ's reign, especially over the thousand-year Kingdom, does not cancel out God the Father or God the Spirit. The rest of the Godhead remains in the background, and Scripture presents God as a triune Spirit Godhead. All three persons continue, not eliminated by Christ's rule.

Les explains that even though Christ rules and reigns, especially during the thousand-year Kingdom reign, the rest of the Godhead is still sort of sitting back there in the background, and we cannot just forget that. He emphasizes that we cannot cancel out God the Father and God the Spirit, because when dealing with God, we are dealing with that triune Spirit Godhead. This comes from 1 Corinthians 15, where Paul says Christ must reign until He has put all enemies under His feet, and the last enemy destroyed is death.

1 Corinthians 15:27

For he hath put all things under his feet. But when he saith all things are put under him, it is manifest that he is excepted, which did put all things under him.

Does Christ's ruling in the millennium mean the Father and the Spirit are set aside?

No. Christ's millennial rule does not set aside the Father or the Spirit. Although Christ reigns during the thousand-year Kingdom, the rest of the Godhead remains in the background and is not canceled. God is a triune Spirit Godhead, so the triune Godhead remains.