We introduce Gillian, a platform for developing symbolic analysis tools for programming languages. Here, we focus on the symbolic execution engine at the heart of Gillian, which is parametric on the memory model of the target language. We give a formal description of the symbolic analysis and a modular implementation that closely follows this descrip- tion. We prove a parametric soundness result, introducing restriction on abstract states, which generalises path condi- tions used in classical symbolic execution. We instantiate Gillian to obtain trusted symbolic testing tools for JavaScript and C, and use these tools to find bugs in real-world code, thus demonstrating the viability of our parametric approach.