The last "assigned seating" trip I was on, the conductor had a list of "open seats" and told us which ones to use as we boarded. However, once in car we found those seats occupied and told to go find other seats, which we did and were never challenged. I suspect BC will be similar. As for offering $s to swap, I wouldn't think that's necessary. Just explain traveling with your parent and I'd bet someone will switch with you (almost always works on planes).

Enjoy the trip.