I actually disagree. Tanks, healers, and DPS are very connected. Perhaps not in the desire to monitor the damage of the tank, or healing potency of the healer, yet it's strongly related none-the-less.
As an example: A "good" DPS can make an under-geared tank, or fresh tank, look like a terrible-tank. Despite that this player is trying their honest-to-goodness-best (eg, a fresh level 50 PLD trying to hold threat off an iLvl 110 BLM doing triple+ flare). Players glancing simply at the direction and target of mobs, and with no understanding of how gear level difference or damage effects enmity, tend to assume that player is simply TRASH.
As another example: In a low-damage party where the mobs aren't going down fast enough, and the tank runs out of cooldowns.. Where the tank makes the assumption that his or her party can handle xxx mass mob based on their HP / a quick gear inspection ... party wipes.
--Who's the first person to get blamed? Usually the tank or healer. EVEN IF they played well. People start chucking things out there like "If you can't tank it, DON'T PULL IT!".
Conversely, maybe people blame the DPS even though everyone's damage was 'fine', and it actually "is" the tank or the healer who's the issue.
Point being that if one person performs poorly and you have no idea whom, it can look like anyone (role independent). Parties are a strongly collective effort where no single role is trivial, and where the problem isn't always obvious / can be assumed to be pretty much anyone. That's for essentially ANY aspect of the group. Additionally someone is always going to be blamed, even if it's just by assumption (that's just human nature sadly).
I'm not going to advocate parsing as I strongly respect SquareEnix's stance on third-party tools if not just for the safety of its playerbase, and I'm certainly not going to advocate people playing the blame-game. Yet I can understand why some people wish to understand what's happening in a fight, and why it's important to know.
There's already some tools ingame that give an idea of this, such as the threat-meter (which can more or less compare "relative" damage between two players of the same class). However ... you'd be surprised how few people actually know of its existence or look at it.