We use first-principles methods to demonstrate that, in ZrTe_{5}, a layered van der Waals material like graphite, atomic displacements corresponding to five of the six zone-center A_{g} (symmetry-preserving) phonon modes can drive a topological transition from a strong to a weak topological insulator with a Dirac semimetal state emerging at the transition, giving rise to a Dirac topology surface in the multidimensional space formed by the A_{g} phonon modes. This implies that the topological transition in ZrTe_{5} can be realized with many different settings of external stimuli capable of penetrating through the phonon-space Dirac surface without breaking the crystallographic symmetry. Furthermore, we predict that domains with effective mass of opposite signs can be created by laser pumping and will host Weyl modes of opposite chirality propagating along the domain boundaries. Studying phonon-space topology surfaces provides a new route to understanding and utilizing the exotic physical properties of ZrTe_{5} and related quantum materials.