Trying to pair my Apple Watch Series 3 to new Iphone 13. Start Pairing and says watch needs update, have went through steps a dozen times and still will not work.

FWIW, a lot of people have problems updating a series 3 watch due to insufficient memory. In some cases, unpairing the watch, completely erasing it, then pairing and restoring the backup will resolve the issue after an update. (Simply unpairing and pairing again may not work.)

Unpair and erase your Apple Watch - Apple Support

Also, see fireyjack's reply to:

Can't update her series 3 watch. - Apple Community