Seems pretty good to me. (I know a little Haskell, but only a little.) I think you and the original author are using different standards of readability. Yours: "If I look at this code, starting without knowing anything about it, does it look familiar?" His: "Once I've taken a few minutes to get the hang of this, is it easy to read and write?".
Here, the <|> and <$> and <* are unfamiliar and weird-looking: fair enough. But once you know that <|> means "or", and <$> means "try to parse some of the input using the thing on the right, and then the thing on the left is the resulting value", and <* means "try to parse both of these, and return the result of parsing the first one", it's hard to see how it could be much simpler.
Even in that code block, <|> and <$> both have roughly the common meanings of | and $ elsewhere in Haskell, which is the "next case" operator for pattern matching expressions and the strictness operator for ensuring that a function is applied to a value correctly.
Here, the <|> and <$> and <* are unfamiliar and weird-looking: fair enough. But once you know that <|> means "or", and <$> means "try to parse some of the input using the thing on the right, and then the thing on the left is the resulting value", and <* means "try to parse both of these, and return the result of parsing the first one", it's hard to see how it could be much simpler.