So I get asked this question a lot and I could go through and recommend the best shoes to buy, but at the end of the day everyone needs to find what shoes work best for them.
Some aspects of shoes are always better based on physics and it's these aspects that you should keep consistent when considering shoes. When training you want grip on your shoes. In order to have grip you need to create friction, so my recommendation here would be brown gum bottom shoes.
A popular base for shoes is foam which is cheap to manufacture but does not grip well to most surfaces. If your shoes have white foam you're probably better off just training barefoot unless you have proper technique to handle the lack of friction. Dyed bottoms are a close second to brown gum but they lose that raw rubber grip and have a small layer of dye that can reduce valuable friction.
Laces, laces, laces. Velcro might be easy for little Johnny to get on his feet, but if the shoe is easy to get on it's just as easy to get off. The weird angles of Parkour landings such as those in the Cat Grab position love to steal peoples shoes right off their feet and leave them hanging at the edge of a roof in their socks. Avoid Velcro at all costs and tie your shoes tight.
Sandles... Really?! Get out. There's no room for you here.
Crocs??? Why do you own those they don't even look good.
Weight of the shoe is an important factor. A heavy shoe can be durable and good for absorbing impact, but detrimental for jump distance. Now we're talking 4-6 inches of distance based on the extra weight you have to drag, but when you're trying to standing jump to a ledge that's over 9ft away every inch counts. This one is preference though. Some people would prefer the extra cushioning for their heavy drops. For example, Vans Ultra Ranges were the shoes I used when I maxed out my standing precision at 9'9" back in 2019. However, the heavier all leather Reebok Classic Workout Plus training sneakers were the shoes I used when I finally overcame my mental block at the Manasquan Inlet. Those shoes felt like armor that protected me.
Thickness of the sole. For Beginners I always recommend a shoe with less padding such as the Onitsuka Tigers Ultimate 81's or Feiyues. These shoes give immediate feedback on how you are performing through the nerve sensors in your feet. A proper landing shouldn't hurt your feet at all. If it hurts you're landing on the wrong part of your foot.
So what shoes should you get for Parkour. Get the right tool for the Job. If you want to flip fast get something light. If you want to better understand your technique get something with light padding. If you want to take big drops, get shoes that will protect you. Start with the cheap shoes that teach you technique until you figure out your style. Then begin exploring which shoes work best for you.