Fair Use in the United States is pretty nuanced and open to interpretation. From section 107 of the Copyright Act:
Courts will balance the purpose and character of the use against the other factors. Additionally, "transformative" uses are more likely to be considered fair. Transformative uses are those that add something new, with a further purpose or different character, and do not substitute for the original use of the work.
Parody is a good example of fair use, though the Copyright Act defines "parody" differently than it defines "satire", and satire is not considered fair use. As I said, it's nuanced.
That being said, Fox most likely paid the copyright owner to guarantee they wouldn't have to defend their use in court (as @Tetsujin said). We'd need to look at a specific instance of reuse from a specific episode of Family Guy.