I just broke up with my girlfriend, and we have met both parents. Shall we send a message to her mother wishing her a happy Mother's Day?

Whether her mother knows you broke up or not, she should send her best wishes. Maybe there will be a chance to make up later. Doing so will give you extra points with her mother, make a good impression on her and think you are a filial child.

So I say you take good care of her mother first, so that your future road will be easier.

You mean her family is against you?