We demonstrate that observations of glitches in the Vela pulsar can be used to investigate the strength of the crust-core coupling in a neutron star, and suggest that recovery from the glitch is dominated by torque exerted by the re-coupling of superfluid components of the core that were decoupled from the crust during the glitch. Assuming that the recoupling is mediated by mutual friction between the superfluid neutrons and the charged components of the core, we use the observed magnitudes and timescales of the shortest timescale components of the recoveries from two recent glitches in the Vela pulsar to infer the fraction of the core that is coupled to the crust during the glitch, and hence spun up by the glitch event. Within the framework of a two-fluid hydrodynamic model of glitches, we analyze whether crustal neutrons alone are sufficient to drive the glitch activity observed in the Vela pulsar. We use two sets of neutron star equations of state (EOSs), both of which span crust and core consistently and cover a range of the slope of the symmetry energy at saturation density $30 < L <120$ MeV. One set produces maximum masses $approx$2.0$M_{odot}$, the second $approx$2.6$M_{odot}$. We also include the effects of entrainment of crustal neutrons by the superfluid lattice. We find that for medium to stiff EOSs, observations imply $>70%$ of the moment of inertia of the core is coupled to the crust during the glitch, though for softer EOSs $Lapprox 30$MeV as little as $5%$ could be coupled. No EOS is able to reproduce the observed glitch activity with crust neutrons alone, but extending the region where superfluid vortices are strongly pinned into the core by densities as little as 0.016fm$^{-3}$ above the crust-core transition density restores agreement with the observed glitch activity.
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