This was discussed a bit on a reddit post way back and I think the popular consensus was:
1) Have the option to turn it on and off in settings
2) Have settings where it be private (practising/learning new class etc) or could be public (raid settings where this actually matters)
3) Have an additional setting that hides public parses from other players (for those that don't care/don't want to see/want to avoid being trolled/etc)
People asking for x or y dps in PFs is about as valid as people asking for specific experience to a phase in PFs. It'll be very apparent once in the instance if the player in question lied/doctored a screenshot/etc.
Most players who actually know the value of a parser will know how dps varies from fight to fight. In terms of raid context the information is crucial, without them if someone isn't pulling their weight and there's no metric for comparison groups could be wiping on progression fights when the fault lies on one person (good groups will usually try to help the player improve versus just exiling him/her off the bat). Other content like dungeons or hm trials don't require parsers for the most part. In the above setting the only people you need to worry about are the trolls (who will still troll via whatever avenue you give them with or without a parser).