When I was graduating from my BA the women wore mortarboards but the men didn’t. When my sister graduated from a college up north everyone wore mortarboards. What’s the standard or does this vary from institution to institution?
I graduated both high school and college in Florida, and everyone wore mortarboards.
It varies by institution. It’s odd that you say you went somewhere where only the women wore mortarboards - the only gendered variation I’ve seen is that the men wear mortarboards and the women wear those soft hat things. I’ve seen it from several universities, but I can’t bring to mind which one right now. (Of course, some institutions have everybody in those soft hats.) Both my undergrad and masters’ degree everybody wore mortarboards.
At St. Andrews the men don’t have mortar boards because, as legend has it, they were so very pissed off at women being allowed in to the university that they all went and threw their mortar boards into the sea by way of protest. (In fact women don’t wear mortar boards either, except when singing in the chapel choir, if I remember rightly. Graduation just involved a gown and a hood.)
Edinburgh doesn’t have mortarboards either. Just a hood and gown, and a pat on the head from John Knox’s breeches 
Although a few people wear them for the photographs after the ceremony.