What Phemie (I believe it was her) described is actually the technically right way of doing it. I’ve been doing weddings for a little over 10 years now in so many different parishes and dioceses. I’m trying to recall if I ever saw it done this way. I think I witnessed it once, but I usually see the “traditional” father walking daughter down the aisle, or both parents or just the bride by herself.
I do know at one of my previous parishes, they did try to implement it, but received such an uproar from the brides and their parents. The brides would be in tears during the rehearsal because it was never explained to them prior. It got pretty ugly, apparently. Pastorally, the pastor of the parish as well as the Director of Worship for the diocese at the time (who lived at the parish) felt that because it was before mass began, it was ok to not insist upon the practice.