I present a novel method for measuring the Hubble constant with pulsar timing arrays, using gravitational waves alone, and without the need for electromagnetic counterparts. Near future arrays could achieve this by simultaneously measuring the luminosity distance and the co-moving distance to supermassive black hole binary sources out to redshifts of z~2. The luminosity distance is measured via the binary chirp, which for pulsar timing arrays appears through heterodyning Earth and pulsar signals. The novel finding here, leading to a Hubble constant determination, is that one can also infer the co-moving source distance from the curvature of gravitational wavefronts, which can be measured from gravitational-wave-timing parallax. In the future, this method could be an important arbiter of other independent measurements of the Hubble constant, which depend on electromagnetic emission models.