First we need to identify the cause of the peeling paint. It's very possible there is an incompatibility between the old and new paint. You didn't mention priming so this is a possibility. Another possibility is the paint isn't fully cured. All paints can take up to 30 days to fully cure.
You could simply wait a week and give the paint some extra time to cure. For the peeled areas, they will have to be sanded smooth and primed with a quick drying oil base primer then repainted.