Traffic question - who was in the wrong - and how to avoid these situations?

@Dutchess’s question reminded me I have a traffic question of my own.

This happened last week. My dad was attempting to merge onto the highway on the right side. He had already accelerated to match the speed of traffic. There was a car in the far right lane next to the merging lane that was mostly behind us but still somewhat “overlapping” us so that he couldn’t change lanes. My dad sped up to try and get ahead of the car, but it sped up too. At this point the merging lane ran out and we were driving on the rumble strip for a moment. My dad had to lay on the horn and the other car moved into the lane to its left. It passed us and their passenger made terrible hand gestures at us.

My mom says we were technically in the wrong because the other car, being already on the highway, had the right of way. But if that’s the case, I’m wondering what would have been the proper course of action in my dad’s situation? It’s not exactly safe to slow down while merging and there may have been other cars behind the problematic one, so I’m not sure that he could have merged behind it.