redshift - How does one measure velocities of far-off, bright objects

The "redshift" of a distant galaxy is defined in terms of its line of sight velocity. In our model of the expanding universe, once we move away from the local group of galaxies (which have their own peculiar motions), distant galaxies follow the Hubble flow and to first order have a line of sight velocity tht is proportional to their distance way (it gets more complicated for very distant galaxies).

Distant galaxies may well have a "tangential" velocity too, but for galaxies outside the local group these velocities will be be negligible compared with the redshift. i.e. The line of sight velocity due to the expansion of the universe is dominant.

I guess by "parallax drift" you actually mean proper motion - which is the rate at which a star's position changes with respect to the celestial coordinate system. This proper motion depends on how far away the star is and how fast it is moving tangentially with respect to the solar system.

Thus to estimate a tangential velocity you need both the proper motion and the distance to the star.

I think the most distant object for which a proper motion has been determined with any accuracy is the Andromeda galaxy, which is a couple of million light years away. This was achieved by studying the position of many stars in Andromeda over a 7 year period using the Hubble Space Telescope. The details can be found in Sohn et al. (2012); but the headline numbers are that the proper motion is a mere $sim 0.05$ milli-arcseconds per year(!) , implying a tangential velocity (with respect to the solar system) of about 150 km/s.

Another candidate is measuring the velocities of material in the jet of the active galaxy M87 by Meyer et al. (2013). This galaxy is at 50 million light years, but the motion of the jet is only detectable here because it is moving relativistically.

These are quite special cases. In general, the tangential velocities of stars in our Galaxy are small and large-scale susrveys of proper motions are generally inaccurate beyond a few thousand light years. The upcoming Gaia results will improve this dramatically meaning we have good proper motions for objects out to tens of thousands of light years.