At a guess, I'd say they allowed the first Winegardner novel to be published simply to test the waters & see if the GF franchise still had legs. Lucasfilm did much the same thing in the early 90s with the Star Wars franchise, although they exercised complete control over new books, comics & video games.

Rumours & talk of 'The Godfather Part IV' has popped up regularly since 1990, Paramount probably figured a new novel wouldn't hurt, & if it was any good &/or successful, they could swoop in & have a head start on a new film.

Given the strong sales of The Godfather Returns, I imagine they thought they might as well let the second novel happen, but obviously the first was received so poorly that it affected the sales of the second.

Now that there's another one on the way, Paramount probably figures they'll be better off cashing in early, that way it doesn't matter if it's a success or not.

Putting aside Paramount's rights to anything with the GF label, just who was Puzo writing the screenplay for? If he was being paid by Paramount at the time to develop a fourth film, I'd have thought Paramount would own it completely, but this doesn't seem so cut & dried.