Sounds like Dad may be nervous or anxious and your daughter can sense that and is responding. Stop taking over when she screams at him cause you are only encouraging a bad habit for your daughter. Not that she "knows" what she is doing but the older she gets the more she WILL know.
Each day, he should spend some one-to-one time with your daughter. Maybe when you go to make dinner or do some laundry in another room. If you start with just a small amount of time at 1st and work up to an hour. You are there for any questions or help he may ask for but don't jump the gun. Let him figure out what's best for THEM. Then when you do have to run an errand, Dad won't feel so anxious and daughter won't get upset.
I once was at a store and walked up to a friend I saw to say hello. A young mother was chatting with her as her baby was screaming. This young mother was doing every thing right to calm her child but it wasn't working. I asked if I could hold the baby (he was maybe 2 months old) so this young mother could at least finish her shopping (I had nothing to do and was bored that day so it kept me from overspending! LOL). It took only mintues for him to stop fussing when I held him. As my friend, this young mother and I, with the baby, walked round the store, the baby had fallen asleep.
Moral of this story? The young mother was overwhelmed, nervous and anxious and I being an "old mom" was not. Even if Dad does not admit to being anxious (most men don't reveal their feelings well! LOL), it sounds like he really is and just needs practice.