How does one measure velocities of far-off, bright objects As the title already says, I want to know how one measures velocities of far-off, bright objects, e.g. when the mean parallax drift isn't measurable with current apparatus (this means when there is only a "non-moving" picture of that object available)

I know that one can measure the redshift of spectral lines and correct them for gravitational redshifting if the distance is already known.
But then the relation between source-frequency $f_S$ and observed frequency $f_O$ is
$$ f_O = f_S \frac{\sqrt{1-\frac{v^2}{c^2}}}{1+\frac{v}{c}\cos(\alpha)}$$,
where $\alpha$ is the angle between line of sight and the velocity vector.

If one only has a "non-moving" picture of the object, there is no way to determine this angle $\alpha$ and therefore no way to get the velocity by means of redshift.

As an addition: At which distance does the mean parallax drift become unmeasurble with current apparatus?
In general the total velocity of a faraway object is not obtainable, only the component along the line of sight. However, if a star is not too far away, and its position on the sky is measured very accurately with an interval of some years, the velocity component perpendicular to the line of sight can be determined, yielding together with the redshift the total velocity. The recently launched Gaia spacecraft is expected to be measure the proper motion of millions of stars in this way.

In distant galaxies, this is not possible. If one is interested in, say, the 3D velocity dispersion of the gas, the 1D dispersion is measured, and assuming (fairly) an isotropic dispersion, this number is then multiplied by $\sqrt{3}$ to get the total velocity.

The distance to which parallax is measurable depends on the angular resolution of your instrument. For milliarcsec resolution, parallax measurements are good out to roughly 1 kpc. If you wait long enough, the motion around the galactic center increases your baseline such that larger distance can be probed.