Translation:I have to do the dishes, but my brother does not have to.
It almost seems to me that 'to be off the hook' is a good English translation for 'å slippe': you were required to so something - or thought you had to do it, but something happened, and now you don't have to do it anymore. I'm going to think of it as 'slipped off the hook.' :-)
'vaske opp' does mean 'wash up' and it can be used in other contexts that just dishes, but that is the most common use. Most other things are either 'vasker' (washing/washes), 'rydder' (cleaning/cleans), or 'rydder opp' (cleans up, tidies, etc.). I generally use 'vasker' for washing things other than dishes, and if I get it wrong, Norwegians usually correct me :)