Edit your edge atribute file to set the edge width to whatever numerical value you want it can be for example the numer of edges linking two given nodes. Then as you import the atributes to the network after removing the duplicated edges create a visual style where you set this property to continuous mapping.
Attribute of edge should be numerical or categorical value. Therefore, if possible, you should change number of edges connecting two nodes to a numerical value which is proportional to the number of edges. Then, using the Edge Line Width property to map to the constructed weights in VizMapper.
I have not much experienced with ver 3.x. Therefore, what I said is about ver 2.x. You can download ver 2.x at http://cytoscape.org/download_old_versions.html
I recomend you read the user guide. There are certainly some improvements on Cytoscape 3.x versions. So, I think you should learn how to do this in the recent version to economize time. Since you will need to use basic features of Cytoscape you wont run into plugins compatibility problems.
In the case of version 3.2.1 is almost the same. Instead Vizmapper use Style. At the bottom of the Style window you will find the toolbar to change the properties of nodes, edges or networks. Go to edges and the rest is very similar to versions 2.x