Does iPhone 13 support Apple Watch Series 10

Yes, when the iPhone is on 18 or newer.

(261979)

If your Apple Watch has already watchOS 26 installed, the iPhone has to have iOS 26 installed as well, which should be no problem with your iPhone 13.

If your iPhone is still using iOS 18, the watch as to run watchOS 11, it can't have watchOS 26 installed to pair with iOS 18 or lower.

Apple Watch and iPhone compatibility – Apple Support (AU)