While helping couples plan their weddings of late, we begin at the entrance of the wedding party, and then the entrance of the bride. Many brides and their fathers are happy for this part of the ceremony. I wonder if they would be just as excited if they knew where that tradition came from. Fathers once used their daughters as currency to: pay off a debt to a wealthier land owner, symbolize a sacrificial offering to an opposing tribe or to buy the families way into a higher social caste. What a barbaric hold over from the days when daughters were nothing but dollar signs.
Luckily there are many options for couples on their weddings these days. I have seen the groom and then the bride walked down the aisle by both parents as a symbol of the family they have come from, or just the bride with her father. In this day and age of blended families, I have seen a step-father escorting the bride halfway and the birth father (depending on the relationship with the bride) escorting the rest of the way. It is not unusual for brides to walk down the aisle on their own. I have even seen a couple come up together.
It is the happy couples day and there is no right or wrong way to enter. Do what ever suits you and have a good laugh over where this tradition came from.