Indeed, the way people go about treating each other gets way too awful in certain duties anyway, even without a parser. Imagine if you saw people's data, it would just help focus attacks on specific members. I do believe that most people know if they're doing enough damage or healing or hold enmity well enough or not. They probably are aware of how they could get better, and try to as well - but they do need practice, it won't get better in the blink of an eye. (And those who aren't aware would probably be hopeless, even with a parser - or maybe they just don't care enough at all, which of course is subject for an entire separate discussion.)
Personally, as I mostly go as DPS (sometimes even when I don't), if I see that the damage is not enough, as it happened with Odin too many times - I just say "we aren't doing enough damage". I include myself in that, even if I suspect I'm doing well enough...for all I know, I might not be. But I don't isolate myself from such statements, as it is a team effort. Even if it's clearly visible to everyone it's because of one specific person. I do believe it takes the edge away, at least somewhat, as opposed to something like "you're doing it wrong". I don't think the atmosphere's ever turned negative when I was the first to call it and did it like that; perhaps it really a minor thing, idk. But maybe worth considering.
(REMOVE THE BLOODY CHARACTER LIMIT ALREADY, FOR SANITY'S SAKE!)