I don't have this book, but it's clearly a typo. Should be:
type Parser a = String -> [(a, String)]
By the way, the author's website has all the code from the book, and if you have a look at Parsing.lhs
, you'll see the definition
newtype Parser a = P (String -> [(a,String)])
That's probably a more advanced version which is obtained incrementally during the chapter, but any way, the fact that a parser has an input string is crucial.