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I'm going to self-post a reasonable solution, in hopes that someone comes up with a better one, which is to use the dist argument in the graph so the edges aren't on top of each other:
G1 = DiGraph(multiedges=True)
G2 = DiGraph(multiedges=True)
var('A,B')
G1.add_edge(A,B,'edge1')
G2.add_edge(A,B,'edge2')
G1.add_edge(A,B,'edge2')
G2.add_edge(A,B,'edge1')
G1.set_pos({A:[0,0],B:[1,0]})
G2.set_pos({A:[0,0],B:[1,0]})
G = G1.graphplot(color_by_label={'edge1':'green','edge2':'white'}, edge_style='--', dist=0.75).plot() + G2.graphplot(color_by_label={'edge2':'blue','edge1':'white'}, edge_style=':').plot()
G.show(axes=False)
