My prescription for sleep-deprived new parents:

•  Tivo. Forget going out to movies, unless there's a crying room. Forget making it through a DVD without passing out. You need Tivo. Time-shifting shows, pausing TV, and skipping commercials ... what more could a parent need? Besides, we're living in the Renaissance era of TV: Arrested Development (RIP), Curb Your Enthusiam, The Wire, West Wing, The Daily Show, The Deadliest Catch (go Ballard!) ... I could go on, but I think I've revealed enough already about how much TV I watch.
  
• Espresso. Nespresso makes a capsule-based home espresso machine that delivers almost instant shots with lovely crema ... all without grinding, tamping, wiping, pulling, and cleaning. As much as I love the auditory baristatory associations of "whirrrr, clunk clunk, flicka-flicka, tap-tap, and psshhhh", for home use at 5am I vote capsule. You can find base-model Nespresso machines refurbished on eBay for $100 or so. Screaming deal.

I'd feel remiss if I didn't mention yoga/exercise and relaxation techniques and eating well too, but that's a different flavor of self-care, and less realistic for folks that aren't getting ANY sleep.