I can’t gibe a full answer but at least I’ll try to give some pointers: You’ll first have to grab the parameter from the URL, pass it to WP and then add the parameter to your links. I’m not aware of any plugin that would do all the three steps out of the box – but there are some fragments available. You would ‘only’ have to put them together and load them as a plugin:
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This (‘abandoned’) plugin could be a starting point, even though it seems not to grab the parameter from the URL. It however seems to provide some sort of ‘link interception’ functionality, adding parameters to ‘all’ the links: YD Spread Parameter
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For the ‘URL grabbing’ part you’ll need a function that a) grabs the parameter AND b) passes it to WordPress. Here is a short tutorial on how to do this: Passing Query String Parameters in WordPress URL