Thanks to Russ, I was able to get this to work correctly, hiding the layer that had editing disabled from the data collection layers in Collector. He told me to "pull to refresh" the map. It's odd that even though I hadn't refreshed the map this way, I had removed it from my iPad and re-downloaded it multiple times, but it still saw the parcel layer as editable.
I also had a line feature service in the map with a null extent. That probably added to the problem.