Commissars I read are like half a rank more senior than the commander of the unit that they are embedded in. Commissars provide a thread for the CCP to manage the military. They are responsible for making it clear to the grunts what they are fighting for, and relaying orders from above and see to it that they are followed. The commander is responsible for the technical side of things i.e. the best way to get a job done or an order accomplished.
The party controls the military, they are not independent. Or more like the military is an extension of the party.
The commissars aren't always more senior in rank. The Chinese rank and grade system are not one-to-one. Each grade, i.e. command position have a minimum rank and a maximum rank, as well as an age limit. The maximum rank of a grade can be higher than the minimum rank of the next grade, but it is the grade that counts in terms giving out orders.
The CO and the commissar have the same grade.
Officers and commissars do not get rotated/assigned at the same time, so their ranks may differ while their grade is the same.
The CO is in charge of tactical and operational decisions, while the commissar is in charge of the administrative part (personal issues, liaisons, overseeing provisions etc). The commissar is also the most senior communist party member in the unit.
Given that usually the CO is also a communist party member, and the notion of the army being an extension of the communist party, this gives a commissar more seniority in practice, especially during peace time. And this is irrespective of the rank.