How can I know the purchase date of my AirPods?

The purchase date will be on the receipt for your purchase.

Look for a receipt: In the packaging / warranty info / if you kept it

In your eMails (search for eMails from Apple)

Wherever you store warranties / guarantees.

etc.

You can check financial statements of the payment method you used.

Apple support's article: Which AirPods do I have? – Apple Support may pinpoint their release date.

Also: You can try entering the serial number of your AirPods pro in Apple Support's webpage (link at the bottom)

How to Find the serial number of your AirPods – Apple Support

You can enter the serial number to see what info. Apple provides:

Check Your Service and Support Coverage - Apple Support