Large Language Models (LLMs) demonstrate increasingly human-like abilities across a wide variety of tasks. In this paper, we investigate whether LLMs like ChatGPT can accurately infer the psychological dispositions of social media users and whether their ability to do so varies across socio-demographic groups. Specifically, we test whether GPT-3.5 and GPT-4 can derive the Big Five personality traits from users' Facebook status updates in a zero-shot learning scenario. Our results show an average correlation of r = .29 (range = [.22, .33]) between LLM-inferred and self-reported trait scores - a level of accuracy that is similar to that of supervised machine learning models specifically trained to infer personality. Our findings also highlight heterogeneity in the accuracy of personality inferences across different age groups and gender categories: predictions were found to be more accurate for women and younger individuals on several traits, suggesting a potential bias stemming from the underlying training data or differences in online self-expression. The ability of LLMs to infer psychological dispositions from user-generated text has the potential to democratize access to cheap and scalable psychometric assessments for both researchers and practitioners. On the one hand, this democratization might facilitate large-scale research of high ecological validity and spark innovation in personalized services. On the other hand, it also raises ethical concerns regarding user privacy and self-determination, highlighting the need for stringent ethical frameworks and regulation.