RenderPartial takes another parameter that is simply a ViewDataDictionary. You're almost there, just call it like this:
Html.RenderPartial(
"ProductImageForm",
image,
new ViewDataDictionary { { "index", index } }
);
Note that this will override the default ViewData that all your other Views have by default. If you are adding anything to ViewData, it will not be in this new dictionary that you're passing to your partial view.