It depends what the issue is.
For my kid with major lifethreatening medical issues, we felt pretty strongly that one parent needed to be really fluent in the medical language and understand the big picture. I quit my job, in part because he needed things like biweekly hospital visits, but also because it allowed me to be that consistent person. So, I went to every appointment, and Dad joined us for the big ones, or the ones when he was off work. Dad worked shifts, so he did have some weekdays off.
For my relatively healthy kids, who are getting more routine care? Then it was usually me, because I was SAH, but if Dad was off he might take them, or if the kids conflicted, he might take a healthy kid while I had their brothers. One of my kids also went through a stage where he was terrified of needles, and his Dad would take him for shot appointments because he did a better job of staying calm with Dad.