With the widespread adoption of multi-core architectures, it is becoming more important to develop software in ways that takes advantage of such parallel architectures. This particularly entails a shift in programming paradigms towards fine-grained, thread-parallel computing. Many parallel programming models have been introduced for targeting such intra-task thread-level parallelism. However, most successful results on traditional multi-core real-time scheduling are focused on sequential programming models. For example, thread-level parallelism is not properly captured into the concept of interference, which is key to many schedulability analysis techniques. Thereby, most interference-based analysis techniques are not directly applicable to parallel programming models. Motivated by this, we extend the notion of interference to capture thread-level parallelism more accurately. We then leverage the proposed notion of parallelism-aware interference to derive efficient EDF schedulability tests that are directly applicable to parallel task models, including DAG models, on multi-core platforms, without knowing an optimal schedule. Our evaluation results indicate that the proposed analysis significantly advances the state-of-the-art in global EDF schedulability analysis for parallel tasks. In particular, we identify that our proposed schedulability tests are adaptive to different degrees of thread-level parallelism and scalable to the number of processors, resulting in substantial improvement of schedulability for parallel tasks on multi-core platforms.