Aims. Relativistic jets launched from active galactic nuclei accelerate up to highly relativistic velocities within a length scale of between a few parsecs and tens of parsecs. The precise way in which this process takes place is still unclear. While magnetic acceleration is known to be able to accelerate relativistic outflows, little attention has been paid to the role of thermal acceleration. The latter has been assumed to act only on compact regions very close to the central engine, and to become negligible on parsec scales. However, this holds under the assumption of small internal energies relative to the magnetic ones, and whether or not this assumption is valid and what happens when we drop this assumption are open questions. Methods. We used a 2D relativistic magnetohydrodynamical code to explore jet acceleration from subparsec to parsec scales. As initial conditions for our models, we used observational constraints on jet properties derived by means of very long-baseline interferometry observations for a Fanaroff Riley I radio galaxy, NGC 315. We investigated the parameter space established for this source and performed a number of simulations of magnetically, thermally, or kinetically dominated jets at injection, and compared our results with the observations. Furthermore, we employed different models to characterize our jets, involving different magnetic field configurations (i.e., force-free vs. nonforce-free) and varying shear layer thicknesses. Results. Our simulated jets show that when thermal energy is comparable to or exceeds magnetic energy, thermal acceleration becomes significant at parsec scales. This result has important consequences, potentially extending the acceleration region far beyond the collimation scales, as thermal acceleration can effectively operate within a conically expanding jet. In all the models, we find acceleration to be driven by expansion, as expected. A number of our models allow us to reproduce the acceleration and opening angles observed in NGC 315. Finally, our results indicate that disk-launched winds might play an important role in jet propagation. Namely, when the jet has an initial force-free magnetic field configuration, thicker shear layers are needed to shield the internal spine from the action of the external medium and thus delay the growth of instabilities.