In this work, we study singlet fission in tetracene para-dimers, covalently linked by a phenyl group. In contrast to most previous studies, we account for the full quantum dynamics of the combined excitonic and vibrational system. For our simulations, we choose a numerically unbiased representation of the molecule's wave function, enabling us to compare with experiments, exhibiting good agreement. Having access to the full wave function allows us to study in detail the post-quench dynamics of the excitons. Here, one of our main findings is the identification of a time scale t0 ≈ 35 fs dominated by coherent dynamics. It is within this time scale that the larger fraction of the singlet fission yield is generated. We also report on a reduced number of phononic modes that play a crucial role in the energy transfer between excitonic and vibrational systems. Notably, the oscillation frequency of these modes coincides with the observed electronic coherence time t0. We extend our investigations by also studying the dependency of the dynamics on the excitonic energy levels that, for instance, can be experimentally tuned by means of the solvent polarity. Here, our findings indicate that the singlet fission yield can be doubled, while the electronic coherence time t0 is mainly unaffected.