Yep, you can run the boards either way. However, running the boards parallel or with the joists means you need to install blocking on 16-inch centers (or less), to match the joist spacing. It should look like a waffle. Deck boards are quite thin and NEED equal and frequent support.
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2x4's might be fine for any person's "load", but I'd go with a minimum of 2x6's as this will help to stiffen the entire deck's structure. Otherwise, it'll look like a pier or dock (purposely perpendicular to the length) and you should be seamless or installing singular 16-foot boards for the best look.