We theoretically investigate the orbital effects of an in-plane magnetic field on the spectrum of a quantum dot embedded in a two-dimensional electron gas (2DEG). We derive an effective two-dimensional Hamiltonian where these effects enter in proportion to the flux penetrating the 2DEG. We quantify the latter in detail for harmonic, triangular, and square potential of the heterostructure. We show how the orbital effects allow one to extract a wealth of information, for example, on the heterostructure interface, the quantum dot size and orientation, and the spin-orbit fields. We illustrate the formalism by extracting this information from recent measured data [L. C. Camenzind et al., arXiv:1804.00162; Nat. Commun. 9, 3454 (2018)].