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We demonstrate how pulsar timing arrays (PTAs) can, in principle, yield a purely gravitational wave (GW) measurement of the luminosity distance and comoving distance to a supermassive black hole binary source, hence providing an estimate of the source redshift and the Hubble constant. The luminosity distance is derived through standard measurement of the chirp mass, which for the slowly evolving binary sources in the PTA band can be found by comparing the frequency of GW-timing residuals at the Earth compared to those at distant pulsars in the array. The comoving distance can be measured from GW-timing parallax caused by the curvature of the GW wavefronts. This can be detected for single sources at the high-frequency end of the PTA band out to Gpc distances with a future PTA containing well-timed pulsars out to $\mathcal{O}(10)$ kpc, when the pulsar distance is constrained to less than a GW wavelength. Such a future PTA, with $\gtrsim 30$ pulsars with precise distance measurements between 1 and 20~kpc, could measure the Hubble constant at the tens of percent level for a single source at $0.1 \lesssim z \lesssim 1.5$. At $z \lesssim 0.1$, the luminosity and comoving distances are too similar to disentangle, unless the fractional error in the luminosity distance measurement is decreased below 10%. At $z\gtrsim 1.5$, this measurement will likely be restricted by a signal-to-noise ratio threshold. Generally, clarification of the different types of cosmological distances that can be probed by PTAs, and their relation to pulsar distance measurements is important for ongoing PTA experiments aimed at detecting and characterizing GWs.