Propositional representation is explained on Wikipedia.
How is semantic parsing different from propositional representation? Or is propositional representation the next step after you do semantic parsing?
Most computational parsing methods yield an output that is a propositional representation within the computer. This includes pure syntactic parsers (if one assumes we can have pure syntax) and semantic and also pragmatic parsers. Why? Because parsers in computers always produce discrete output and every discrete output can be converted into a graph representation.
Distributional representations of words (word2vec) would be a representation that is not propositional. But such a representation is (currently) not used as the main structural component of a parser output.
If you think about semantic parsing in the human it is a different story and I would not say that it is clear what kind of processes or representations semantic parsing includes in the human brain.
Both entail the compositional process of converting natural language into a formal or machine-interpretable language. I believe the difference is that propositional representation is how some psychologists/psycholinguists believe people interpret speech, as a series of discrete symbols much like a formal language. This is related to the idea of 'mentalese', if you're familiar with that. Semantic parsing, on the other hand, is a deliberate process which is carried out to make natural languages into formal meaning representations which can be interpreted by machines. This is a computational idea and has nothing to do with psychology.