Discovering symbolic representations for skills is essential for abstract reasoning and efficient planning in robotics. Previous neuro-symbolic robotic studies mostly focused on discovering perceptual symbolic categories given a pre-defined action repertoire and generating plans with given action symbols. A truly developmental robotic system, on the other hand, should be able to discover all the abstractions required for the planning system with minimal human intervention. In this study, we propose a novel system that is designed to discover symbolic action primitives along with perceptual symbols autonomously. Our system is based on an encoder-decoder structure that takes object and action information as input and predicts the generated effect. To efficiently explore the vast continuous action parameter space, we introduce a Curiosity-Based exploration module that selects the most informative actions -- the ones that maximize the entropy in the predicted effect distribution. The discovered symbolic action primitives are then used to make plans using a symbolic tree search strategy in single- and double-object manipulation tasks. We compare our model with two baselines that use different exploration strategies in different experiments. The results show that our approach can learn a diverse set of symbolic action primitives, which are effective for generating plans in order to achieve given manipulation goals.