This book should probably have been called "Rincewind's Return" and not Eric. It is neither about Faust (in any of the versions) nor about the not-quite-14-year-old would be demonologist, Eric Thursley. As Tony has noted, TP had a busy year and perhaps the work with Neil Gaiman on Good Omens had some minor influence. Certainly the sections in Hell seem to me to be the funniest. But the most significant feature, I think, of this very slim and only mildly funny novel is the use of the three wishes--an element which seems to be totally absent from all the Dr. Faustus versions.
The most significant characteristics of the three wishes folk legend, which Terry will use later in Hat Full of Sky , are that the wishes usually work quite literally, though not in the way expected, and that the third wish has to be used to undo the harm of the first two. And to some extent Eric's wishes follow the ususal pattern.
Traditionally in the folk tales, there is no concern for the "happiness" of anyone other than the individual with the witches. Terry, however, adds the idea that the three wishes are designed to make the greatest number of people happy. But it's not entirely clear what happened to Eric. Certainly his experiences about being a ruler and having a chest of gold, turn out to be illusory. His wish to live forever is not sufficiently detailed though. And of course, Helen is a middle-aged mother by the time he meets her.
I agree, the parrot is totally unnecessary as well as unfunny. There are some small elements of humor, especially when they finally arrive at Hell. But the rest of the book is not particularly funny. And it shows, in my opinion, the speed at which is must have been written. There are a fair number of "loose ends" and some outright contradictions.
For example, Rincewind (feeling that things could have been a lot worse), steps off the road of good intentions, through a wall which healed up behind him. As far as I can tell, Eric is left standing on "For the Sake of the Children," and one suspects that Eric may not be happy.
I have to say that I didn't hate the book--which I did the first time I read it. I did find parts of it mildly amusing, especially Pratchett's depiction of the new version of Hell. I haven't seen the illustrations, but as I dislike Kirby's work, I doubt this would improve my feeling about the book.
All of this puts me in the unusual position of joining Pooh and Quatermas in thinking this is a mighty thin book.