@daisy83 When I was the working parent I got up for nighttime feedings and to help out. We did the shift thing and that worked really well for us.
It sucks, for sure, being tired at work but 1) it also sucks be tired, at home, with kids and 2) its hard work all day long so the idea that resting at night somehow makes the day different never really made sense to me.
Also... I don't know anyone's specific situation but I work a high visibility, high impact, very difficult job and never felt like I was going to be fired for being tired at work. I think the whole "Well I have the job so I need my rest or else we will starve" thing is a bit overblown, if you want it you can make it work.