The definition of the scattering volume for p-wave collisions needs to be generalized in the presence of dipolar interactions for which the potential decreases with the interparticle separation as 1/R3. Here, we propose a way to define the scattering volume characterizing the short-range interactions in odd-parity waves, by analyzing the p-wave component of the two-body threshold wave function. Our approach uses an asymptotic model and introduces explicitly the anisotropic dipole-dipole interaction, which governs the ultracold collision dynamics at long range. The short-range interactions, which are essential to describe threshold resonances, are taken into account by a single parameter related to the nodal structure of the wave functions at short distances.