The Beatles wrote it. If they hadn't written it, The Carpenters would not have ever recorded it BECAUSE THE SONG WOULD NOT EXIST. Even if they'd never recorded anything; The Beatles would be famous as songwriters.
And you are wrong about which version is better (though I do like Karen's version as well).
Speakig of their songwriting prowess, “Yesterday” is the most covered song by other artists of all of them.