As COVID-19 ravages the world, social media analytics could augment traditional surveys in assessing how the pandemic evolves and capturing consumer chatter that could help healthcare agencies in addressing it. This typically involves mining disclosure events that mention testing positive for the disease or discussions surrounding perceptions and beliefs in preventative or treatment options. The 2020 shared task on COVID-19 event extraction (conducted as part of the W-NUT workshop during the EMNLP conference) introduced a new Twitter dataset for benchmarking event extraction from COVID-19 tweets. In this paper, we cast the problem of event extraction as extractive question answering using recent advances in continuous prompting in language models. On the shared task test dataset, our approach leads to over 5% absolute micro-averaged F1-score improvement over prior best results, across all COVID-19 event slots. Our ablation study shows that continuous prompts have a major impact on the eventual performance.