Is the Apple Watch series 3 compatible with the iPhone 13

Yes. But FWIW a lot of people have had trouble updating a series 3 watch. With the limited memory in a series 3 watch, your best option if you have trouble pairing the series 3 watch h with an iPhone running iOS 15 may be to make an appointment at the Genius Bar in an Apple Store near you and let them look at it:

Genius Bar Reservation and Apple Support Options - Apple

See the replies to:

Apple Watch 3 is stuck on preparing updat… - Apple Community

The apple watch 3 GPS only has only 8GB of internal storage, most of that is taken

up by the operating system and other critical software with not much space left for the

update.So installing or even pairing to an ios15 phone is proving nearly

impossible for just about all the watch 3 users on this forum.

same applies to Apple watch 1 and 2 they are almost impossible to either pair or

update with an ios 14 or ios15 phone.

Your best course of action is to call in at an apple store and ask them to pair/update it for you.

Yes