This paper presents an automata-based algorithm for answering the /emph{provenance-aware} regular path queries (RPQs) over RDF graphs on the Semantic Web. The provenance-aware RPQs can explain why pairs of nodes in the classical semantics appear in the result of an RPQ. We implement a parallel version of the automata-based algorithm using the Pregel framework Giraph to efficiently evaluate provenance-aware RPQs on large RDF graphs. The experimental results show that our algorithms are effective and efficient to answer provenance-aware RPQs on large real-world RDF graphs.