Several task models have been introduced in the literature to describe the intrinsic parallelism of real-time activities, including fork/join, synchronous parallel, DAG-based, etc. Although schedulability tests and resource augmentation bounds have been derived for these task models in the context of multicore systems, they are still too pessimistic to describe the execution flow of parallel tasks characterized by multiple (and nested) conditional statements, where it is hard to decide which execution path to select for modeling the worst-case scenario. To overcome this problem, this paper proposes a task model that integrates control flow information by considering conditional parallel tasks (cp-tasks) represented by DAGs with both precedence and conditional edges. For this task model, a set of meaningful parameters are identified and computed by efficient algorithms and a response-time analysis is presented for different scheduling policies. Experimental results are finally reported to evaluate the efficiency of the proposed schedulability tests and their performance with respect to classic tests based on both conditional and non-conditional existing approaches.