The option settings False and None are not listed among documented typical settings, but both MultiedgeStyle -> False and MultiedgeStyle -> None give the desired result.
GraphPlot[{1 -> 2, 2 -> 1}, MultiedgeStyle -> False, DirectedEdges -> True, VertexLabeling -> True]

GraphPlot[{1 -> 2, 2 -> 1, 1 -> 3, 3 -> 2}, MultiedgeStyle -> False, DirectedEdges -> True, VertexLabeling -> True]

GraphPlot[{1 -> 2, 2 -> 1, 1 -> 3, 3 -> 2}, MultiedgeStyle -> None, DirectedEdges -> True, VertexLabeling -> True]

Notes:
- This works in version 9 (windows 10) and in version 11.2 on Wolfram Cloud.
- After major updates in version 12.0,
GraphPlot usage/options changed completely. Use GraphComputation`GraphPlotLegacy instead of GraphPlot for the code above to work.