The main issue with adding a DPS parser to the game is that it focuses the game on DPS.

I agree with this issue.
But this is not a problem with adding a parser.
This is a problem with adding a parser in the wrong way.

A DPS-only parser gives the impression of things being DPS-focused and it can destroy a community by providing a lopsided view of what's required to be a good player.

A DPS-only parser would be terrible in FF14 simply because DPS doesn't tell the whole story about being a good player. It just shows if you can skew one facet of the game to lean more your way for a certain fight. Exceptionally high DPS in a fight only shows you relied on your other party members to do mechanics or to heal you through what you should have spent time on dodging.

So, what's the good way?
Well, what makes a good player?

A good player (in no particular order):
-Manages their DPS (more, more, more!)
-Manages agro (either enough as a tank or not-too-much as a non-tank)
-Manages their health (either by healing or dodging avoidable AoE)
-Manages mechanics (helping the group clear)

Therefore, a good way to do parsing would be to have a metric for each of these important aspects.

Having "just a DPS parser" is as good as having "just an agro parser."
Both provide useful aspects on their own, but both are laughable as some sort of evaluation for "how you are as a whole player." This only sows confusion into the community. The kind of confusion that breeds negativity.

So, show people how they can actually be "good players."
Give us a good parser.
Show us what we all want to know.

Example of a good parser:
-provide an understandable DPS metric
AND -provide an agro metric
AND -provide a healing metric
AND -provide a damage taken metric
AND -provide a mechanics-followed metric

If you do all of this, then you can do "a parser" in the right way and add a great level of benefit to the entire community.

No one can say "your DPS is low" if right beside that value it shows you did all the mechanics in the dungeon and no one else did any. Without you, everyone else would have 0 DPS.

No one can say "learn to heal" when the metric shows they stood in the last 5 avoidable AoEs.

Worried about trolling?
Easy fix to that.
Keep the same regulations about reporting players who use numbers to abuse others.
Make the metrics non-sharable-in-game (personal parsers). You can even only have them show up after you leave the instance.
Make the metrics available (showable) as an option. The default is off. Players must want to find the setting and turn it on.

There are many, many ways to do a good parser that shows all areas of how a "good player" should be. It would be great to know how we're doing and compare it to the average-last-50-runs of the same instance or something like that.

I fully agree that we should not add "just a DPS parser." But, I really, really, really want a good parser that shows all the aspects of playing this game well.