This opinion should surprise no one who knows me or reads this blog. I'm all for replacing Mother's/Father's Days with one Parent's Day, and I'm about to tell you why!
1) Binary gender ideals are a lie. Even if it miffs you, please realize that an inclusive society needs to divorce itself from rewarding some gender identities over others. If religion and state should be separate, so should gender and state. I acknowledge Mother's Day is a private affair, not a public one, but democratic public institutions are where a Parent's Day would be created/legitimated.
2) It takes a village. I have a step-Mom who played a huge role in my life, and I have seen first hand how Mother's Day can be a little awkward. Think about the children with even more complex parental arrangements in communitarian settings, foster care, and multi-generational homes.
3) Social expectations of motherhood shame childless women. Motherhood is put on a pedestal as the epitome of femininity and selflessness. It's not. Mother Theresa, Tyra Banks, and Oprah chose not to have children. Further, mothers are expected not only to love their children but to love motherhood. If women's autonomy is to actually be respected, we need to be able to talk about women who regret motherhood without stigmatizing their legitimate feelings and experiences.
4) Mother's Day is a Hallmark holiday. Consumer holidays don't improve relationships. Time is better than stuff, and mainstream media will berate you with suggestions to the contrary. I teased out my full thoughts on this matter in a December post on Post-Materialist Gifting.