Parenting is exhausting work, especially when you have a newborn in the house; you have to be available when they need something though, so you learn to stumble through and get the job done, no matter how tired you are.
Luckily, taking care of your baby becomes such second nature that you can soon do it on auto-pilot when you have to. When your baby starts crying at 2 AM, you manage to feel your way through the dark (with your eyes still mostly closed) and get halfway through the feeding before you even realize you're out of bed. You can do drowsy diaper changes, sleepily sway with them in your arms, or instinctively snag the pacifier and wipe it off before you consciously notice she dropped it.
In short, parenting becomes such second nature that you can't turn it off.
Even when you're sleeping, apparently.