From what I can tell, there are many reasons why pizza operators use screens (and disks as well). A big reason--the one you mentioned--is that it is easier to load a pizza into an oven on a screen than from a peel and you also don't need to use cornmeal or any other release agent. Also, if you'd like, you can "deck" the pizza from the screen onto the stone to get the desired degree of bottom crust browning. Screens can also be used to keep the bottom crust soft, by raising the pizza above the deck surface by the thickness of the screen. If the dough has a lot of sugar in it, the screen will also serve to keep the bottom crust from browning prematurely. In a home setting, it is not necessary when using a screen to preheat the oven for an hour or more as you would with a pizza stone or tiles.
Once in a while you will hear of a pizza operator using parchment paper in an oven, but it its very uncommon and most likely used to address a unique problem with an oven. Most of the time, parchment paper is used by pizza operators either as separators for a stack of skins or for take-and-bake applications. I tried using parchment paper for some take-and-bake pizzas that I experimented with in my home oven and found that it did a good job. I was able to bake the pizza directly on an oven rack at normal oven temperatures (e.g., around 425 degrees F), with good results. In fact, the results were better than I thought they would be, with better bottom crust browning than I expected. But parchment paper for my use can be quite expensive, far more so than using a release agent on a peel or using a screen over time. I have also not been able to locate a source of parchment paper with dimensions big enough to allow me to make a 16-inch or larger pizza. Otherwise I would be able to use it on a large screen (as a carrier) or on my 20-inch wood peel to bake pizzas on an oven rack at normal oven temperatures. One negative that I have noticed with parchment paper is that it can disintegrate under baking, creating its own mess to clean up either in the oven or as it is being removed from the oven.