I think I can offer a unique perspective. I am the full-time working parent. DH has been SAHD and now primary homeschooling parent from the beginning. I am also a recovering control freak.
I am the researcher in the family. It's just what I do. It's what I do for a good part of my job too so I can't help but do it. I research things like crazy, but I talk to DH about it. I talk to DD. We're really in this together.
DD really prefers to have us lead her education to some extent. We have bought some curriculum to try to meet her interests. So, I don't think we're as unschooly as most. But, we work on plans together, and we have goals as a family. DD wants to be a stronger reader so we work on that together, for example.
I realized at some point that I would buy all these cool books that I would hope DH would share with her during their days, and it just wasn't happening. He had missed the conversation DD and I had about something, and just being 6, she didn't always remember what books I had told her about when she wanted to learn more later on. We basically had a real disconnect.
So, we're trying different ways to make sure DH knows what resources he has available. We have a wiki page set up now where I put everything I find. We have a folder on our home server where we share materials. We plan more together.
We've also divided up our focal interests. I work with language arts (reading, writing, spelling), art, and Spanish. He works on math and science (though I am former scientist.. I just don't have time during the day to do science with DD.. we read a lot about science). I say this is pretty loosely divided, though. It's more like we're the resident "go to" people on these topics.
I do have the benefit of being a full-time telecommuter so I have some flexibility, but I work for a huge corp and have to keep a basic corporate schedule and I have tons of meetings. But, I dedicate a couple hours every morning to just work with her. We read aloud, practice reading, work on Spanish, do something social sciencey (she loves learning about other cultures and religions). It's pretty much whatever we feel like doing in the mornings, but I do wrap it around these general topics.
DH and DD work on other things however they've chosen to do so for the day/week. He goes to all her activities with her--homeschool field trips, classes, etc. They do daily errands together, etc.
In the evenings, DD and I will often do something arts/crafty together before dinner, and after dinner we all usually play games together.
We've managed to achieve a balance that we're all pretty happy with right now. I still get to do the research, etc. that I love--they both tell me things they want to explore, I suggest things too, and then I help them find the resources. I am able to still be an active part of her schooling/living/learning too even though I work. There are some days, though, where they are so busy doing things outside the house that I feel like it's no different than if she were in school. I have to remind myself that is my perspective not hers...she is with her dad doing homeschool things, and that is much different than being in school.