How can I know my iPhone 13 Pro purchase date?

You can check the date the warranty was activated. That will tell you the date it was originally sold.

Check Your Service and Support Coverage - Apple Support

If you bought the phone from Apple or an Authorized Apple Reseller, that should be the same as the date you bought it.

Look at the receipt you got when you purchased it.

If you bought it second hand, ask whoever you got it from.