Parser generators and parser combinator libraries are the most popular tools for producing parsers. Parser combinators use the host language to provide reusable components in the form of higher-order functions with parsers as parameters. Very few parser generators support this kind of reuse through abstraction and even fewer generate parsers that are as modular and reusable as the parts of the grammar for which they are produced. This paper presents a strategy for generating modular, reusable and complete top-down parsers from syntax descriptions with parameterized nonterminals, based on the FUN-GLL variant of the GLL algorithm. The strategy is discussed and demonstrated as a novel back-end for the Happy parser generator. Happy grammars can contain `parameterized nonterminals' in which parameters abstract over grammar symbols, granting an abstraction mechanism to define reusable grammar operators. However, the existing Happy back-ends do not deliver on the full potential of parameterized nonterminals as parameterized nonterminals cannot be reused across grammars. Moreover, the parser generation process may fail to terminate or may result in exponentially large parsers generated in an exponential amount of time. The GLL back-end presented in this paper implements parameterized nonterminals successfully by generating higher-order functions that resemble parser combinators, inheriting all the advantages of top-down parsing. The back-end is capable of generating parsers for the full class of context-free grammars, generates parsers in linear time and generates parsers that find all derivations of the input string. To our knowledge, the presented GLL back-end makes Happy the first parser generator that combines all these features. This paper describes the translation procedure of the GLL back-end and compares it to the LALR and GLR back-ends of Happy in several experiments.