To take your example there, I would say the one on the parallel taxiway has the right of way.
The landing aircraft has been given a landing clearance, which lets him be on the runway.
The guy taxiing on the parallel taxiway has a clearance to be on that taxiway.
If the landing traffic wants to go on to the taxiway, he should obtain a taxiclearance from ATC.
If there are high speed exit taxiways at the airport you are refering to, I can go as far as to say that the landing traffic can land and vacate onto one of the high speed exit taxiways, but not taxi further than that, and to hold short of the parallel taxiway (which should be just after crossing the holding point on that taxiway.)