You don't want to expose to YOUR parents unless he has to leave the home. They will likely never forgive your H and that may cement him simply wanting to escape the shame.
Your call, but in laws are painful enough to deal with let alone having to walk the gauntlet of having cheated on thier daughter.
Hold off on that unless you absoutely have to.
OR, when he leaves you tell HIM to do it and hand in the phone so he does it honestly.