IMO, none of the discussed "pieces" is the "missing piece." Harris, Super Mario, and Parker won't put us over the Warriors or Rockets.
In the cases of Super Mario and Parker, not only NO but HELL NO.
Super Mario is Rodney Hood 2.0. He does ok when he's playing against inferior competition and his jumper is going in. But when faced against tough matchups or isn't getting the ball, he falters rather than rises. He's one dimensional, just like Hood. If he's scoring, he's effective. But when he's not scoring, he doesn't contribute in any other way.
Parker is more injury prone than Exum and doesn't defend. Due to his religious affiliation, there will be a large segment of the Jazz fan base that will put pressure on the franchise to make Parker a focal point of the team rather than a role player off the bench.
IMO, outside of Leonard or James coming to Utah for free, there isn't a FA out there that significantly upgrades this team. The upgrades need to come through current player development and improvement and smart draft choices. Royce O'Neal must be a focus. If he can improve his 3 point shot, much like Ingles has done in his career, he could be James Posey/Shawn Marion Lite, a perfect player to start alongside Mitchell. Crowder needs to lose about 10-15 lbs to improve his quickness.
That doesn't mean we shouldn't go after any FAs. But I don't think we should break the bank for anyone either. I like Patrick Beverly and Terry Rozier. One thing that stood out to me after Rubio went down was our lack of guards who could create off the dribble for others. If we could somehow pick off Beverly or Rozier, I think that upgrades our team in a spot of great need.
Does it put us over the Warriors or Rockets? Probably not.
But these teams are super teams. Outside of signing another big time start like Lebron James or Kawhi Leonard, there's no way we are going to beat a healthy Warriors team who start arguably the best PG, SG, and SF in the entire league.
Perhaps the best way to beat them is to wait them out? Golden State is getting older and Houston has got to start losing players, right? And how much longer can CP3 keep this up? If the Jazz play their cards right, Gobert and Mitchell will be in their primes just as Houston and Golden State are in decline.