Hi Richard,
In V 12.1 you can use EdgeTaggedGraph.
g = EdgeTaggedGraph[{Style["A" \[DirectedEdge] "B", Green], 
   Style["A" \[DirectedEdge] "B", Cyan], 
   Style["A" \[DirectedEdge] "B", Blue]},
  VertexStyle -> Black,
  GraphLayout -> "LayeredDigraphEmbedding",
  ImageSize -> 48]
