Can I pair my Series 3 Watch with my iPhone 14?

You can pair it like any other combination. I was able to get a Series 2 paired with an iPhone 11 with relative ease. See this article:

Set up your Apple Watch - Apple Support

Note that some things will differ since your Watch is likely running an older version of watchOS, not 10 (the latest), but it should be the same process.

Pairing an Apple Watch 3 to a newer iPhone can be a real pain to get paired. But if you’re lucky you can get it done