Does the father of the bride give a gift to the mother of the bride? What did you do?

I can't say I've ever heard of this.
My wife and I did give a special gift to our sons wife on their wedding day.
Our daughter in laws parents are both deceased, her dad killed on a auto accident when she was 3 and her mother passed from heath issues when she was 12. She was passed around to relatives, met my son in high school and basically hung out at my house her jr and sr years of high school.
If it hadn't been for that circumstance, we wouldn't have done what we done.