DH doesn't consider himself to be "babysitting" the kids - he's a very involved father (when he's home!).
However, he does still get a mindset where I'm the primary caregiver e.g. he will often assume that he is free to do things without checking with me - he just figures I'll be with the kids. Or, when I joined yoga and dance classes, it seemed like a huge hassle to arrange for him to be home from work on time, and when he was away I needed to organize childcare. I just don't feel as "free" as he seems to be, and I find that frustrating. He has a demanding job and a demanding hobby, and that adds up to a lot of time that he's not with us. If I were to plan activities for myself that took up as much time as his hobby (fishing) does, we'd never see each other - we'd spend our lives handing off the kids between each other. It doesn't seem fair sometimes, but it's not worth ending a marriage over - I still love him, even though I do find this one thing difficult. I suppose no spouse is perfect!