I think it was right of her to step down simply because so many people wouldn't understand that the tweets she posted were three years old, and that she didn't necessarily share the same viewpoints now. She made a legitimate point in that she felt that pressure from the media would have prevented her from fulfilling her duties in that post.
That being said, I do think that the tweets she posted that were picked up and highlighted by the media would have alienated Paris from other teenagers who were from the groups she was essentially slagging off (LGBT, ethnic minorities, etc). The media cunningly neglected to mention the age of her tweets, so for all they knew, those were her current thoughts. Would you want to speak to someone from the police if you knew they were openly against your sexuality or race?
The issue with giving any youngster a position in the police service is that it is such a public-centric role that you really do have to watch what you say everywhere. This isn't something that the majority of people understand - if you work for the police, you will be under scrutiny, end of. You can't afford to have lapses of judgement and to post the odd questionable remark here and there, you just can't.
I can appreciate what the Commissioner was trying to achieve by creating this role in the first place, but I'm not convinced it's the smartest of ideas given the age group they're targeting.