Im 6ft3 and pushing 400 pounds ive never had a problem on BA flights. Yes i need a seatbelt extension but only so I will be comfortable. I admittedly do fit between the armrests. 

I do feel bad for people next to me because my shoulders are quite broad but i try to get an aisle seat so i can at least lean out. 

Since your mother is travelling with your father it should be fine. They will even be able to put he armrest up if she needs more space.