Parker is averaging 2.1 assists per game for his career
Barnes is averaging just 1.5
That is a pretty negligible difference if that's your argument for him being better. Complimentary does not equal playmaking, and that really doesn't say much for his playmaking anyway.
Because he demands the ball a few more assists will naturally come off having more touches. Barnes does not demand the ball, he does his job.
His defense is atrocious and advanced stats will tell you he is arguably a detriment over a plus on his teams.
Raw offense is all he's got over Barnes right now. I watch watch way too much basketball tbh, and guys like him and Jordan Clarkson are flashy...but they do not play in the modern approach to the game. Washington might be the right landing spot for him though. He might finally become quite the effective player if he can actually get fully healthy with ample playing time off the bench, and not have to worry about a starting/lockdown role.