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]