Plex grammars according to [13], generating two-dimensional plex structures, are a generalization of string grammars. In this paper we describe a parser for context free plex grammars. The parser is an extension of Earley's algorithm, which was originally developed for context free string grammars. Our parser is able to recognize not only complete structures generated by a plex grammar but also partial ones. The algorithm has been implemented and tested on a number of examples. The time complexity of the parser is exponential in general, but there exist subclasses of plex languages for which the parser has a polynomial time complexity.