Regular path queries are a way of declaratively expressing queries on graphs as regularexpressionlike patterns that are matched against paths in the graph. There are two kinds of queries: existential queries, which specify properties about individual paths, and universal queries, which specify properties about all paths. They provide a simple and convenient framework for expressing program analyses as queries on graph representations of programs, for expressing verification (model-checking) problems as queries on transition systems, for querying semi-structured data, etc. Parametric regular path queries extend the patterns with variables, called parameters, which significantly increase the expressiveness by allowing additional information along single or multiple paths to be captured and related. This paper shows how a variety of program analysis and model-checking problems can be expressed easily and succinctly using parametric regular path queries. The paper describes the specification, design, analysis, and implementation of algorithms and data structures for efficiently solving existential and universal parametric regular path queries. Major contributions include the first complete algorithm and data structures for directly and efficiently solving universal parametric regular path queries, detailed complexity analysis of algorithms for solving existential and universal queries, detailed analytical and experimental performance comparison of variations of the algorithms and data structures, and investigation of efficiency tradeoffs between different formulations of queries.