Wouldn't Parker meet himself when he backstepped, thus effectively making two of him? To this troper's knowledge, meeting himself in the past was never an issue nor was it explained away.
We're led to believe that the instant a Parker (and the Sphere) arrive from the future, the version currently existing disappears into thin air, as a correlary to the "a person can't exist in the same place twice" theory of time travel. Thus the instant the Sphere disappears, they just wait for Parker to call.
This is backed up in one episode where the Sphere leaves without Parker. We skip to the past where Donovan and Ramsey are going "yeah, wonder where Parker is this time", and are surprised when Parker appears right behind them, eating chips, asking what's going on. He doesn't know, since he's never at the base when they do the "sphere's gone, something bad happened" thing.
It's interesting that to them each launch is a first launch/they Never actually do a launch. Because when they send him back, that changes the timeline so where they don't need to send him back, so don't do a launch. He fixes the problem that needed to be fixed so they don't launch him. But eventually a new problem comes up that they need to launch him back, but to them, it is the first time they ever launch him because him coming back prevents the problem so they don't need to launch him........ Only Parker experiences ALL the launches. To everyone else they only know about the arrivals.
On the flip side, they're not completely blind. I'm sure they've done a million launch rehearsals. The fact that the Sphere keeps disappearing, and Frank keeps making Conundrum calls with future information proves that whatever they're doing is working. Frank can later describe how each launch went from his perspective, and from there they can make tweaks. They don't get to see each launch, but if they can trust Frank to stop terrorist plots, they can trust Frank when he says the backstep went off without any problem.