Psalm 127:2 describes sleep as a divine gift: "for so he giveth his beloved sleep." The Hebrew word for "beloved" — yedidaw — is the same word used for David in the Psalms, meaning the cherished one, the one in particular favor. Sleep is not a concession God makes to human weakness. It is what he gives to those he loves. The new parent who catches sleep in fragments is not failing to trust God adequately. They are receiving, in interrupted form, what God gives to those he cherishes.
Mark 6:31 records Jesus pulling his disciples away from active ministry explicitly for rest: "Come ye yourselves apart into a desert place, and rest a while: for there were many coming and going, and they had no leisure so much as to eat." The need around them was real. People were still arriving. Jesus withdrew them anyway. Rest was not incidental to the mission. It was how the mission continued.
Commentary is from a charismatic Protestant perspective, drawing on KJV text and public-domain sources including Spurgeon, Andrew Murray, and Matthew Henry.