We use a newly developed observing mode on the Hubble Space Telescope (HST) and Wide Field Camera 3, spatial scanning, to increase source sampling a thousand-fold and measure changes in source positions to a precision of 20-40 μas, more than an order of magnitude better than attainable in pointed observations. This observing mode can usefully measure the parallaxes of bright stars at distances of up to 5 kpc, a factor of 10 farther than achieved thus far with HST. The technique should also provide a unique crosscheck of future parallax measurements from Gaia. Long-period classical Cepheid variable stars in the Milky Way, nearly all of which reside beyond 1 kpc, are especially compelling targets for parallax measurements from scanning, as they may be used to anchor a determination of the Hubble constant to ∼1%. We illustrate the method by measuring to high precision the parallax of a classical Cepheid, SY Aurigae, at a distance of more than 2 kpc, using five epochs of spatial-scan data obtained at intervals of six months. Rapid spatial scans also enable photometric measurements of bright Milky Way Cepheids—which would otherwise saturate even in the shortest possible pointed observations—on the same flux scale as extragalactic Cepheids, which is a necessity for reducing a leading source of systematic error in the Hubble constant. We demonstrate this capability with photometric measurements of SY Aur on the same system used for Cepheids in Type Ia supernova host galaxies. While the technique and results presented here are preliminary, an ongoing program with HST is collecting such parallax measurements for another 18 Cepheids to produce a better anchor for the distance scale. [ABSTRACT FROM AUTHOR]