When you decide to start a new generation and there is still a parent, he/she gets "kicked out" which means they disappear and the picture on the family tree will stay lit up. It only dims when you witness a person's death in the game.

I didn't know about this and accidentally booted out my generation 1 female (who I named after myself) So I made up a story that she lived the rest of her life in a luxury nursing home in Florida. It's all up to you though! It's so hard seeing them pass away and deal with the loneliness and depression.. but I do feel better from letting them live it through and get happy again.