We had full names on the table plan as had a few David's and the likes, but used "known as" names (including nicknames) on the place cards. Where there were multimple David's on the table they had the first initial of their surname on too. TBH, it wouldn't have mattered too much, i'm sure they would have worked out which seat to sit on. And you're not gonna care on the day if they sit the wrong side.
As for parents - it was also known as names, so Mam / Dad for my parents and as he has step-parents it was Mum & Dad and then first names for the step parents as that's what he calls them (so Jane and Fred - he doesn't call them Step-Mum or Step-Dad). Again, Dad's could work out where they should sit and even if they somehow swapped over, it wouldn't have mattered to me.