The following multigraph plots correctly (i.e. parallel edges do not overlap) using graphviz neato to generate a png (as shown in this answer)
import network
Well I know its probably not what you're looking for, but I was facing a similar problem where I wanted to have a directed graph where the edge between two nodes had a different weight depending on the direction (whether it was going into or out of the node) and the work around I did was I used a different color for each edge and decreased the opacity for one of them so it would show even if they overlap. I only needed two edges between my two nodes so it did the trick for me.
G = nx.DiGraph()
G.add_nodes_from([0,1])
pos = nx.circular_layout(G)
nx.draw_networkx_nodes(G, pos, node_color = 'r', node_size = 100, alpha = 1)
nx.draw_networkx_edges(G, pos, edgelist = [(0,1)], width = 2, alpha = 0.5, edge_color='b')
nx.draw_networkx_edges(G, pos, edgelist= [(1,0)], width = 1, alpha = 1)
plt.axis('off')
plt.show()