The short, rushed answer is that the two concepts are mutually exclusive, but it will also depend if you're talking about pre-exiting parallel realities existing side by side or if you're talking about different realities within the same timeline branching out because of a time-traveler's meddling. In the former, with each parallel reality having its own independent, closed-loop timeline, then I guess you could have a grandfather paradox in each one, but these different realities wouldn't have anything in common with each other, and thus wouldn't be "parallel" to begin with (as they won't share a timeline at any point in their history). In the latter, the paradox wouldn't exist as every change in the past would take you to a new modified timeline-reality.
So I'd say if you want different worlds all having a grandfather paradox, you technically could, but they wouldn't be "parallel" realities. Each reality would be its own independent bubble of events. And since they won't share anything with other realities, you would have to create the paradox in each one, individually, and they would all be different, and each reality would need a time-traveler born and raised there to commit the act. If you create the paradox in one reality, it will be self-contained and will not ripple out into other realities. If they don't share the timeline, they don't share the paradox.
So to clarify, you need to be clear if you want to travel to other pre-existing realities/dimensions (a paradox in each universe would be possible but they share nothing in common with each other and are not really parallel), or if you want to travel within parallel versions of the same universe ( which is what I think you were originally asking about. No paradoxes would be possible, and they share their history until point of divergence caused by the traveler). That's my take on it.
Also, check out the self-consistency principle.