It's just common sense that in order to keep peace in your household, chores should be split evenly between the people who live there. Whether you share chores with roommates, a partner or both, this two prong approach will get them to do more chores without a fight.
Image from emutree.
The first prong is to take a look at the chores that each person is supposed to do and evaluate if they're similar in effort or time needed to complete them. As behavioural economist Dan Ariely points out, try to be objective and not assume the chores you do are harder or require more effort just because you don't see all the tasks the other person handles. If you both agree the chores seem unbalanced, switch them around so they feel fair. The way my boyfriend and I do this is by alternating the chores we handle each day, with Sunday as a combined effort. We both end up doing each task the same number of times each week.
The second prong is to show gratitude to the other person for the effort they put into keeping the household running. This is easy to forget to do over time, but The Greater Good lays out why it is important, particularly with relationships:
In one set of studies, researchers asked romantic partners about their household duties, how appreciated they felt for completing them, and how satisfied they were in their relationships. The team found that the more people felt appreciated by their partners for the chores they did, the more they reported wanting to do them and even liking to do them. For most people, doing more chores meant they were less satisfied with their relationships, but this negative effect evaporated for people who felt appreciated for the chores they did.
So genuinely thanking the other person for taking care of their chores helps change the dynamic around them into something positive. Besides this two part approach, you can also try some tricks to make doing chores more fun, like making it time you spend together blasting your favourite playlist or racing each other (or the clock) to get your tasks done.