Linked Stream Data extends the Linked Data paradigm to dynamic data sources. It enables the integration and joint processing of heterogeneous stream data with quasi-static data from the Linked Data Cloud in near-real-time. Several Linked Stream Data processing engines exist but their scalability still needs to be in improved in terms of (static and dynamic) data sizes, number of concurrent queries, stream update frequencies, etc. So far, none of them supports parallel processing in the Cloud, i.e., elastic load profiles in a hosted environment. To remedy these limitations, this paper presents an approach for elastically parallelizing the continuous execution of queries over Linked Stream Data. For this, we have developed novel, highly efficient, and scalable parallel algorithms for continuous query operators. Our approach and algorithms are implemented in our CQELS Cloud system and we present extensive evaluations of their superior performance on Amazon EC2 demonstrating their high scalability and excellent elasticity in a real deployment.