Two become one in the marital act. Notice this is specific to becoming “one flesh”, not “one person” or “one spirit” or “one mind”, etc.
You are still two distinct people with distinct opinions, feelings, etc.
We are not commanded to kick our parents to the curb when we marry (in fact, remember how Jesus reacted to those who were not caring for their parents?).
The hierarchy of relationships changes, if your mom is having a stroke and at the same time you are supposed to meet your wife for lunch, the most important thing is to get mom to a hospital.
I will be honest, I have had exactly one disagreement with my in laws over decades of marriage, my husband has never had a disagreement with my parents. My adult child’s partner and I have never had so much as a tiny spat. I have to stretch my mind to imagine squabbling over how much time is spent with other people vs spent with me.
My advice, remind your parents that right now, the newborn and your wife need all of the attention and time you can spare. Ask them to stop arguing with your wife and ask them to please make up for the falling out. The ball is in their court, they are adults. Tell them that “the ball is in your court, mom and dad. Do not force me to choose between you and my spouse. Apologize and let’s all move on.”