For flows with a conformal hyperbolic set, we establish a conditional variational principle for the dimension spectra of Holder continuous functions. We consider simultaneously Birkhoff averages into the future and into the past. We emphasize that the description of the spectra is not a consequence of the existing results for Birkhoff averages into the future (or into the past). The main difficulty is that even though the local product structure is bi-Lipschitz, the level sets of the Birkhoff averages are never compact. Our proof is based on the use of Markov systems and is inspired in earlier arguments in the case of discrete time.