My parts don't show (my avatar is a braidout done on wet hair) and from what I can see in your siggy, I think it's very possible for you to get your parts hidden well. We all know wet hair braidouts shrink more, but they also give a certain fatness to the hair when they air dry. I've got really thick hair and I'm natural too. I think it's a combination of not using too much product (it makes the hair skinnier), separating the braids and fluffing the hair that helps me hide the parts easily. I will do braidouts on dry hair too, but I still dampen the hair slightly because my hair gets way too flattened with a dry hair braidout and no moisture.
I don't just massage the roots, I fluff my hair, whether it's dry or damp at take down time, because it's usually MOSTLY dry and that's good enough (I'm just more gentle in this case). I don't try to hold on to that perfect definition because my hair's not long enough yet to just take the braids down and shake. My braidouts usually turn out very pretty though. I get more hang with the ones that are done on dry hair and more fullness from the ones done on wet hair and I'm able to hide the parts either way by doing what I mentioned above.
HTH!