Interesting exercise:
Sit down together with a blank sheet of A4 paper in front of both of you.
Draw two lines the length of the page, landscape, dividing the page into long thirds.
Each line is your typical average weekday. Draw 14 little marks down the page and label them 8am to 10pm.
Mark off on each line what each of you do - to make this interesting, try to fill out each other's lines. He tries to fill out your line, you try to fill out his line. Ask each other if you don't know. E.g. don't just fill out 8 hours as "Work" or "Looking after the kids" for either of you, instead ask what they generally do at what time at work, if anything. What does "Looking after the kids" actually mean? Get detailed!
Don't just include "jobs" here either. Include lunch breaks, time spent watching TV, time spent travelling etc etc.
By the end of all this, both timelines should be pretty full of writing.
Now, this is the important bit:
Does either timeline have manifestly, obviously, hugely disproportionate zones of stuff that isn't actually "productive"?
If so, do either of you see this as a problem?
Secondly, look at weekends, and think about what happens there in the same way. Maybe you don't need to write it down, because I guess you might do more stuff together then.
My point is that if one of you works, and the other one doesn't, it can be very easy to see the other as having it easy. For example, if I look after the kids all day, and my partner comes home, and falls over on the sofa instead of cooking dinner, then it's easy to think "Oh, why am I always doing this, it's so unfair" but it could easily be that I've had quite a lot of spare time during the day e.g. during the kids naps, when I'm not having to directly supervise them at soft play, when we go round to someone's house for a chat and some tea. By comparison my partner has been in meetings for four hours, teaching for two hours, and preparing presentations for an hour, with a 30 minute lunch break and a 45 minute drive at each end, so it's hard to begrudge them the chance to fall over at the end of all that.
None of which is to say that anyone is automatically entitled to anything - if either person isn't pulling their weight, that can easily be a problem. My point is that it's easy to perceive someone as slacking, if you don't actually have a sit down chat, in detail, as to what it is that they do every day. The partner who works may also find a new appreciation for the difficulty of looking after the kids once one of you makes a 20 point list about what that actually entails.
Hopefully it'd at least give you the chance to walk in each other's shoes, and serve as a good way to springboard into a discussion about how much time anyone should be spending doing what.
Good luck!