Is there a published algorithm for translating a context-free parsing problem into SAT? That is, an algorithm that translates a context-free grammar and an input string into a set clauses that is satisfiable iff the input string is well-formed according to the grammar.
Translation of context-free parsing into SAT
fl.formal languages;sat
(I guess the important word in the original question is ``published''.)  There is such an encoding of context-free parsing (more exactly of CYK-style parsing) in  Roland Axelsson, Keijo Heljanko, and Martin Lange, Analyzing Context-Free Grammars Using an Incremental SAT Solver, ICALP 2008, Lecture Notes in Computer Science vol. 5126, pp. 410--422, doi:10.1007/978-3-540-70583-3_34.  They use it in particular to detect ambiguity of words $w$ in context-free grammars for growing word lengths.