  We have observed a dramatic change in the spectrum of the formerly heavily
absorbed `overlapping-trough' iron low-ionization broad absorption line
(FeLoBAL) quasar FBQS J1408+3054. Over a time span of between 0.6 to 5
rest-frame years, the Mg II trough outflowing at 12,000 km/s decreased in
equivalent width by a factor of two and the Fe II troughs at the same velocity
disappeared. The most likely explanation for the variability is that a
structure in the BAL outflow moved out of our line of sight to the ultraviolet
continuum emitting region of the quasar's accretion disk. Given the size of
that region, this structure must have a transverse velocity of between 2600
km/s and 22,000 km/s. In the context of a simple outflow model, we show that
this BAL structure is located between approximately 5800 and 46,000
Schwarzschild radii from the black hole. That distance corresponds to 1.7 to 14
pc, 11 to 88 times farther from the black hole than the H-beta broad-line
region. The high velocities and the parsec-scale distance for at least this one
FeLoBAL outflow mean that not all FeLoBAL outflows can be associated with
galaxy-scale outflows in ultraluminous infrared galaxies transitioning to
unobscured quasars. The change of FBQS J1408+3054 from an FeLoBAL to a LoBAL
quasar also means that if (some) FeLoBAL quasars have multiwavelength
properties which distinguish them from HiBAL quasars, then some LoBAL quasars
will share those properties. Finally, we extend previous work on how
multiple-epoch spectroscopy of BAL and non-BAL quasars can be used to constrain
the average lifetime of BAL episodes (currently >60 rest-frame years at 90%
confidence).
