I'm a dad of two kids, but married to a man, so we didn't have preconceived gender roles. Instead of looking at what is fair division, look at what's important to you and your partner. Then take that role on.
For me, birthday cakes, homemade is important. He prefers store bought. I do that. For him, laundry should be done when the basket is full, for me, it should be done when the cupboard is empty. He does that.
The issue is if one parent is doing too much or things don't get done. The other stuff that is not important, you just have to divvy up. It's important that you're both clear on that. I wouldn't to leave the cake to him and be disappointed it's storming bought. If he's busy and can't do laundry, he tells me. Obviously both help out where possible. If I'm at end of closet and going a load of clothes, I do his and the kids. If he's at the shop the day before a kids birthday, he checks it I'm baking or need ingredients.
You just need to cover the basics. The rest is as it happens. However, when simple stuff like school forms. Having one parent responsible for all permission slips means less checking with each other as to if it's done. One parent can just ignore emails and reminders.