The second way, which I think would look really neat, is to put all parallels of each individual card in a single sleeve. So, the base and the 7 parallels that I'm collecting in a single page. I think this would add some really neat pizazz to the display, and make each page look a bit trippy.
I do this with my ND alum football stuff, I like having the parallel progression immediately visible but I am faced with empty spots on occasion. I also don't put one player per page just due to how few cards some have. That 16 card top loader binder seems like a winner because I'm looking at revamping my PC the same way you are.
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