SetEdgeWeights

We create a graph with multiple parallel edges.

h  =  GraphSum[g  =  Cycle[4],  g]

ShowGraph[h]

?SetEdgeWeights

Here we set edge weights usin the function SetEdgeWeights. This will assign random edge weights (in the range [0,1]) to the edges. Note that all copies of {1, 2} have the same edge weight. SetEdgeWeights does not have an option that allows different copies of an edge to have different weights.

t = SetEdgeWeights[h]

Edges[t,  All]