Embth, welcome to the forum and an early congrats on your WFO. I started pizza making with parchment paper just as you did and then eventually graduated up to a wooden peel, and a GI metal/perforated peel. I currently use both wooden and the GI metal perforated peel.
What peel you need will in part depend on several factors. 1) what style of pizza you want to make 2) hydration which has been mention already 3) more importantly how well the gluten is developed in the dough 4) how much and what you use as the bench and peel flour 5) how long the pizza sits on the peel prior to launching 6) and techniques to unstick the pizza should it happen.
A lot of the above factors have already been mention so I'll just touch up on the ones that haven't been.
1) For most NY/NH style pies, thin crust, America/California styles of pizza you should probably stick with a wooden peel. You may have seen those metal perforated peels made by GI metal. They are quite popular on the forum and really meant for the Neapolitan style. I'm not saying you can't use the peels interchangeably for different styles but the peels do fit their respective styles better.
2) and 3) This has already been mention but when the dough strength is built up sufficient, the dough will not stick. If at the time of opening your dough, the dough opens up too easily and is too sticky or just too difficult to work with, this is a dough that has not been developed properly relative to the hydration level. The problem could be that the hydration is just too high for that strength of flour or it hasn't been developed properly through kneading, folding, or balling.
4) Not using enough bench flour to cover the entire surface of the dough ball or not flouring your peel can also lead to some sticking issues, but this usually isn't the culprit. It's usually 2) or 3). As far as how much bench flour to use, the dough ball should be covered entirely or immersed into bench flour and then the excess shaken off as the dough is opened. As mentioned, corn meal, rice flour, etc on the peel also helps but may not be necessary, unless you like the flavor and texture that cornmeal adds to the bottom crust. I usually just use AP or 00 to cover the dough ball and peel.
5) When using the metal perforated peel, the pizza is meant to be built off the peel, then dragged onto the peel, quickly restretched and loaded. You can also build the pie on a smooth surface and quickly slip the peel under the pie as well. The pies are not meant to be built on the peel as you would do with the wooden peels. I did have sticking issues with the metal perforated peel, but it was due to either building the pie on the peel ( a no no), not building sufficient strength into the dough, or not using any peel flour.
Generally speaking you don't want the pizza sitting on the peel any longer than is required to quickly sauce, cheese, top, and load. The longer a pizza sits, the higher the chance of it sticking to the peel.
6) There are a couple of methods you can also use to help prevent sticking. Some people will blow some air under the pizza before loading the pie as a visual cue that the pizza is not stuck. I don't use this method but have seen it used in a commercial setting before. Some will say that it is gross and unappetizing, but be assured any nasties will be baked off.
Also mentioned already, you can give the pizza a slight shake back and forth before loading as well.
Now, back in the day when I started and had a major sticking issue at the last minute. As a last resort in salvaging a pie, I have covered the pie with a stiff board, flipped the covered pizza and peel over to get the pizza off, refloured the peel, and reflip. Thank goodness I haven't had to do that but once or twice.
You can also make calzones as mentioned above if you screw up a pizza and don't want to waste it.
And finally, rice flour and a wooden peel will work just fine. Good luck.