Being a stepmom is the hardest job I've ever had
Another very wise woman reminded me that as a step-parent, your role is to "love them, cherish them, and know when to keep your mouth shut". Basically saying that though you're a parental figure of sorts, you're not the parent, and need to leave the parenting to Mom & Dad.
This is much easier said than done...and it becomes even more difficult when there is animosity between the two parents. It's during these situations that it becomes even more important for the step-parent to know when to stay in the background and keep out of it. Even when Mom (as an example) is being a complete hound to Dad, and is about to claim the title of Worst Co-parent Of The Year, and your instinct is to get right up there and fight on behalf of Dad...you can't. You just can't. It's very difficult to have to silence that urge, but you really have to just - basically - put up and shut up.
It's also critical that the step-parent never criticizes the other parent - ever. Not even a whispered curse (and trust me, we've all been
there, too

). While it's really the parent's job to facilitate the relationship between the child and the other parent, the reality is that when there's bitterness and anger, the step-parent is often the one reminding Dad (in your case for example) , and the child, that Mom loves kiddo no matter what, and just because she and Dad might be angry with each other, BOTH parents only want what's best for the child.
If you can do that with a genuine smile on your face, you're doing a grand job!
(heck, you can get t-shirts made up with "Team Kid" on 'em - for ALL of you. It's a simple gesture, but it reinforces the idea that this isn't about who's on Mom's or Dad's side, and that it's not about "fighting"...but that all of you are on kiddo's side and just want what's best. Presenting one to Mom is also one heck of an olive branch, showing that Dad just wants to work WITH her, not against her)
(I'm rambling again, aren't I?)