One of the best things about parkour is that you don't need much expensive equipment. Since the point is efficient movement through the environment, you want as little between you and your environment as possible. If you are just getting started, all you really need is a good pair of shoes. Some other types of gear, such as sports holsters and gloves, can come in handy for protection and convenience.
Easily the most important piece of equipment you'll want for practicing parkour is a good pair of sneakers. When looking for shoes, you want to find a model that provides enough padding to cushion hard falls and jumps. Grip is also extremely important, since you will be moving fast over varied obstacles. Lastly, your parkour shoes need to be durable. Cheap shoes will fall apart quickly, which can increase your risk of injury.
Some experienced traceurs actually practice barefoot or in minimalist shoes, as this can provide better feedback when navigating the environment. Unless you have developed the strength in your feet to go barefoot, stick to good sneakers.
In terms of clothing, wear something you are comfortable in and that won't restrict your movement. For most traceurs, loose shorts and a T-shirt work fine, but some people prefer more closely fitted clothes. Your clothing should be light and breathable, since you will be working up a sweat and moving around a lot.
Things to Carry
Always carry some form of ID. If you do get into an accident while practicing parkour, you need people to be able to identify you if you need medical help. Keys and bulky phones can get in the way or get crushed if you happen to land on them, so you should keep a small bag or backpack nearby for stashing your wallet, phone and keys. Some traceurs use a sports holster, which is a sling with pockets across your shoulders. These are designed to stay out of the way during rolls, but they are too small for carrying much beyond the essentials.
You don't need a lot of protective gear in parkour because wearing it can encourage sloppy technique. Depending on the environment or on your own needs and preferences, however, you might want to wear gloves. They should be fingerless to improve your grip. Keep in mind that gloves aren't very effective in wet conditions, and they will prevent your hands from developing protective calluses.