Accurate modelling of the conductivity structure of mineralisations can often be difficult. In order to remedy this, a parametric approach is often used. We have developed a parametric thin-sheet code, with a variable overburden. The code is capable of performing inversions of time-domain airborne electromagnetic data, and it has been tested successfully on both synthetic data and field data. The code implements an integral solution containing one or more conductive sheets, buried in a half-space with a laterally varying conductive overburden. This implementation increases the area of applicability compared to, for example, codes operating in free space, but it comes with a significant increase in computational cost. To minimise the cost, the code is parallelised using OpenMP and heavily optimised, which means that inversions of field data can be performed in hours on multiprocessor desktop computers. The code models the full system transfer function of the electromagnetic system, including variable flight height. The code is demonstrated with a synthetic example imitating a mineralisation buried underneath a conductive meadow. As a field example, the Valen mineral deposit, which is a graphite mineral deposit located in a variable overburden, is successfully inverted. Our results match well with previous models of the deposit; however, our predicted sheet remains inconclusive. These examples collectively demonstrate the effectiveness of our thin-sheet code.