This paper generalizes the notion of a strong LL(k) context-free grammar for the case of conjunctive grammars and develops a top-down parsing algorithm for the resulting language family. A top-down parser of a conjunctive language attempts to construct a derivation of the input string, at each step using a finite lookahead to determine which grammar rule to apply. Fragments of formulae that form the derivation are stored in a tree-structured pushdown. Two ways to implement top-down parsers are suggested: a table-driven parser with a tree-structured pushdown and a recursive descent parser. Both techniques naturally extend their context-free counterparts.