|How do you directly or indirectly encourage your kids to listen to the parent who is not at home all day?
Firstly, I encourage the children themselves to connect with thier dad, even if they have to work at it. He is by nature not a very "connectable" person.
I do understand that it's hard, especially with dh's who have certain personality traits. There came a point that I realized we were going to need to actively work on this, and dh realized he was soon going to lose his children's hearts if things didn't change. Since my dh is not abusive, and doesn't actually require anything outrageous or impossible from the children (even though I tend to be "softer" and more lenient than he), I back him up 100%. I have also explained some of dh's quirks (moody, depressed, and his very high expectations) to the children and taught them the best way to respond when dh is down in the dumps. I've had to work hardest with our oldest child, as he and dh didn't get off to a very good start.
An example--today I needed to stay home from church with the baby, we two being sick. Dh was glad to take the older two boys with him. I knew that our oldest (5) would whine and fuss about missing me etc, even though he wanted to go to church, and I knew that would annoy dh and put a wedge in their relationship, and that the two of them would just feed of each other's irritation the whole day. So I first talked with ds about his attitude, reminded him about how miserable everyone gets when one person starts whining and crying, and asked if he had any particular concerns about going with dh (He did--he was worried that dh would forget to take him to Sunday school!). I then presented ds's concern to my husband at breakfast in a non-accusatory way, explained ds's fear of trying to find his way around church himself, and dh said it would be just fine if ds reminded him when it was time to go to Sunday school, and he would take the boys himself. Barring some unforseen circumstance, I expect that we have headed off any possible head-butting between dh and the older boys at church, and hope to greet them with the door and see their smiles very soon.
My playing the mediator is fairly common and works for us. Also, no matter what we're doing or how much it would screw up our schedule, any time dh volunteers for time with the kids, I go for it. I also try to make sure we eat meals together as much as possible, and that we are home when he is home, so we all have as much exposure to each other as possible.