Yeah, definitely remove all the parts.

If it's plastic, you'll want a dye like Krylon Fusion.
If it's metal, you'll want to sand the current paint off, use a layer of primer, sand to smooth, spray a layer, sand (repeat the paint/sand until smooth), and then use a clear coat.

Would I be correct in thinking you want to paint the base chassis? If so, you'll really need to tear the system and case apart if you want to do it correctly.