My mom loves me just as much as my future wife will. And my mom dedicated her life to my brothers and I as we grew up. Shouldn't my girlfriend get permission from my mom, or at least tell her our plans before I propose?
Oh wait here's how it should work: I ask my girlfriend's dad if I can propose. If he says yes I propose to her. She says "hold on" and goes and asks my mom if she can accept my proposal. She responds in accordance with what my mom told her.
If this is really about gaining the respect of your future inlaws and not rooted in an unequal view of women and their ability to make their own decisions; then the girlfriend should ask my parents too. Her father wouldn't be happy with her marrying just anyone and my mom feels the same about me.
I really do think all the parents should know, but it should start with each person telling their own parents that there is talk of getting married. And no one should ask permission.