Maybe someone can help me out. I've always wondered what it takes to actually receive songwriting credits. For example, Black Francis is credited with writing most of the Pixies songs. Does that mean, then, that he wrote the lyrics, guitar, bass, and drum parts? If not, then why aren't the Pixies credited with writing the songs instead?

It means that he wrote the lyrics and the "music", i.e., the melody. Songwriting credits usually have nothing to do with the individual instrumental parts, even in jazz where the musicians are often improvising around a central theme.

Writing credits can certainly be confusing. Just look at the Beatles. All of John and Paul's songs are credited to Lennon/McCartney even though it was common knowledge even at the time that the two almost always wrote separately. And a lot of bands take a collective songwriting credit even when one or two members do the bulk of the writing. But generally the rule is that if you contributed to the music or the lyrics of a given song, you're entitled to a songwriting credit.