Neither is right and they are going to need to compromise. (Which is sounds like the father is trying to do.)
They need to work out a detailed order and knowing that the child's birthday and Father's Day will fall on the same day some years, need to iron that out in there too.
My birthday falls right before Labor Day weekend, every time. Sometimes is ON Labor Day weekend. Knowing this, I worked out a compromise that settled the issue before it became an issue. My ex agreed to it. He still sometimes forgot and would pick up our kids on what would normally be his Tuesday or Thursday because he forgot that my birthday was either THAT day or had been the previous weekend when he had them for Labor Day and I was to get them for the following Tuesday.